Contact three-manifolds with exactly two simple Reeb orbits

It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegenerate. Combined with a previous result, this implies that the three-manifold is diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are the core circles of a genus one Heegaard splitting. We also obtain further information about the Reeb dynamics and the contact structure. For example the Reeb flow has a disk-like global surface of section and so its dynamics are described by a pseudorotation; the contact struture is universally tight; and in the case of the three-sphere, the contact volume and the periods and rotation numbers of the simple Reeb orbits satisfy the same relations as for an irrational ellipsoid.

Geometry & Topology msp Volume 27 (2023) Contact three-manifolds with exactly two simple Reeb orbits 1 Introduction

Statement of results
Let Y be a closed oriented three-manifold.Recall that a contact form on Y is a 1-form on Y such that ^d > 0. A contact form has an associated Reeb vector field R defined by the equations simple if the map is an embedding.Every Reeb orbit is the k-fold cover of a simple Reeb orbit for some positive integer k.
The three-dimensional case of the Weinstein conjecture, which was proved in full generality by Taubes [42], asserts that a contact form on a closed three-manifold has at least one Reeb orbit; see [28] for a survey.It was further shown in [11] that a contact form on a closed three-manifold has at least two simple Reeb orbits.This lower bound is the best possible without further hypotheses: Example 1.1 Recall that if Y is a compact hypersurface in R 4 D C 2 which is "starshaped" (transverse to the radial vector field), then the standard Liouville form .x i dy i y i dx i / restricts to a contact form on Y .If Y is the three-dimensional ellipsoid and if a=b is irrational, then there are exactly two simple Reeb orbits, corresponding to the circles in Y where z 2 D 0 and z 1 D 0, with periods a and b, respectively.
One can also take quotients of the above irrational ellipsoids by finite cyclic group actions to obtain contact forms on lens spaces with exactly two simple Reeb orbits.
It is conjectured that, in fact, every contact form on a closed connected three-manifold has either two or infinitely many simple Reeb orbits.This was proved by Colin, Dehornoy and Rechtman [8] for contact forms that are nondegenerate (see the definition below), extending a result of [12].It was also shown by Irie [35] that, for a C 1 -generic contact form on a closed three-manifold, there are infinitely many simple Reeb orbits, and moreover their images are dense in the three-manifold.
The goal of this paper is to give detailed information about the "exceptional" case of contact forms on a closed three-manifold with exactly two simple Reeb orbits.
To state the first result, let D Ker. / denote the contact structure determined by .This is a rank-2 vector bundle with a linear symplectic form d .If W R=T Z !Y is a Reeb orbit, then the derivative of the time T flow of R restricts to a symplectic linear map Daniel Cristofaro-Gardiner, Umberto Hryniewicz, Michael Hutchings and Hui Liu We also obtain additional dynamical information.To state the result, recall that the contact volume of .Y; / is defined by vol.Y; / WD Z Y ^d : Theorem 1.5 Let Y be a lens space and let be a contact form on Y with exactly two simple Reeb orbits, 1 and 2 .Then: (a) Let p D j 1 .Y /j < 1, let T i 2 R denote the period of i , and let i 2 R denote the "Seifert rotation number" of i ; see Definition 4.3.Then (b) is dynamically convex, and the contact structure D Ker. / is universally tight. 2   Example 1.6 For the ellipsoid in Example 1.1, we have T 1 D a, T 2 D b, 1 D a=b, 2 D b=a, p D 1, and vol D ab.Thus Theorem 1.5(a) implies that if Y D S 3 , then the periods T i , the rotation numbers i , and the contact volume satisfy the same relations as for an ellipsoid.For Y D S 3 , under the additional assumptions that is nondegenerate and is the standard contact structure, it was previously shown by Bourgeois, Cieliebak and Ekholm [5] and Gürel [19] that "action-index relations" hold, implying that the periods T i and rotation numbers i satisfy the same relations as for an ellipsoid.The equation vol D T 1 T 2 that we prove in this case answers [4, Question 2].
Remark 1.7 There exist contact forms on S 3 with exactly two simple Reeb orbits which are not strictly contactomorphic to ellipsoids.One way to see this is to start from Katok's construction [37] of Finsler metrics on S 2 with exactly two closed geodesics, such that the Liouville measure on the unit tangent bundle is ergodic for the geodesic flow.Such a geodesic flow can then be lifted to a Reeb flow on the standard contact 3-sphere with the same properties.Another way to see this is by Albers, Geiges and Zehmisch [1], who showed that the pseudorotations from Fayad and Katok [15] can be realized as the return map on a disk-like global surface of section for a Reeb flow on the standard contact 3-sphere with precisely two periodic orbits; see Section 1.3 below.On the other hand, Helmut Hofer has suggested to the authors in private 2 Recall that a contact form on a three-manifold Y with c 1 ./j 2 .Y / D 0 is called dynamically convex if CZ. / 3 for every contractible Reeb orbit , where CZ denotes the Conley-Zehnder index (see Section 2.2) computed with respect to a trivialization which extends over a disc bounded by .A contact structure on Y is universally tight if its pullback to the universal cover of Y is tight.
Geometry & Topology, Volume 27 (2023) Contact three-manifolds with exactly two simple Reeb orbits correspondence (2021) that perhaps imposing the additional condition that the rotation numbers of the two Reeb orbits are Diophantine forces the contact form to be strictly contactomorphic to an ellipsoid; see Fayad and Krikorian [16,Question 6].
Remark 1.8 As shown by Honda [23,Proposition 5.1] (see Cornwell [10, page 17] for more explanation), each lens space has either one or two universally tight contact structures up to isotopy, and when there are two they are contactomorphic (and one is obtained from the other by reversing its orientation).Consequently, in Theorem 1.5(b), the contact structure is contactomorphic to a "standard" contact structure on the lens space obtained as in Example 1.1.In particular, universally tight contact structures on lens spaces are precisely the ones that admit contact forms with exactly two simple Reeb orbits.Some other results obtaining information about contact structures from Reeb dynamics can be found in work by Etnyre and Ghrist [14], Hofer, Wysocki and Zehnder [20; 22], and [24].
Remark 1.9 We also obtain information about the knot types of the simple Reeb orbits 1 and 2 .It follows from the Heegaard splitting in Corollary 1.3 that these are p-unknotted.We further show in Section 5 that their self-linking number is 1 when p D 1; similar arguments show that, for general p, their rational self-linking number, as defined by Baker and Etnyre [2], equals 1=p.

Outline of the proofs
We now briefly describe the proofs of Theorems 1.2 and 1.5.
A key ingredient in these proofs, as well as in the related papers [11; 12], is the "volume property" in embedded contact homology, which was proved in [13].The embedded contact homology (ECH) of .Y; / is the homology of a chain complex which is built out of Reeb orbits, and whose differential counts (mostly) embedded pseudoholomorphic curves in R Y ; see the lecture notes [30] and the review in Section 2. The version of the volume property that we will use here asserts that if Y is a closed connected 3-manifold with a contact form , then Here f k g is a "U -sequence" in ECH, and c k is a "spectral invariant" associated to k , which is the total symplectic action of a certain finite set of Reeb orbits determined by k ; these notions are reviewed in Section 2.
The outline of the proof of Theorem 1.2 is as follows.Let 1 and 2 denote the two simple Reeb orbits, and let T 1 and T 2 denote their periods.Simple applications of the volume property from [11; 12] (just using the k 1=2 growth rate of the spectral invariants and not the exact relation with contact volume) show that the homology classes OE i 2 H 1 .Y / are torsion, and the ratio T 1 =T 2 is irrational.A more precise use of the volume property then gives the relations where i 2 R is the Seifert rotation number that appears in Theorem 1.5(a), while `. 1 ; 2 / 2 Q is the linking number of 1 and 2 ; see Definition 4.2.The proof of (1-3) also depends on a new estimate for the behavior of the ECH index (the grading on the ECH chain complex) under perturbations of possibly degenerate contact forms, which is proved in Section 3.
The equations (1-3) imply the relations Since `. 1 ; 2 / is rational and T 1 =T 2 is irrational, it follows that 1 and 2 are irrational.This implies that 1 and 2 are irrationally elliptic (see Section 4.4), which completes the proof of Theorem 1.2.
The Heegaard decomposition in Corollary 1.3 implies that `. 1 ; 2 / D 1=p, and combined with (1)(2)(3)(4) this proves Theorem 1.5(a).The proof of Theorem 1.5(b) uses additional calculations in Section 5 to deduce dynamical convexity and universal tightness from information about the numbers i .

Pseudorotations
The contact forms studied here are analogous to "pseudorotations", defined in various ways as maps in some class with the minimum number of periodic orbits.For example, according to Ginzburg and Gürel [18] a Hamiltonian pseudorotation of CP n is defined to be a Hamiltonian symplectomorphism of CP n with n C 1 fixed points and no other periodic points; see eg Le Roux and Seyfaddini [39], Shelukhin [41] and Çineli, Ginzburg and Gürel [7] for generalizations to other symplectic manifolds.More classically, we consider here pseudorotations of the open or closed disk defined as area-preserving homeomorphisms with one fixed point and no other periodic points; see eg Bramham [6] and Fayad and Katok [15].
In fact, there is a direct connection between the contact forms considered here and pseudorotations of the closed disk.Recall that given a closed three-manifold Y with a contact form , a disk-like global surface of section for the Reeb flow is an immersed disk, with boundary on a Reeb orbit, embedded and transverse to the Reeb flow in the interior, such that the Reeb flow starting at any point in Y hits the disk both forwards and backwards in time.
Corollary We thank the referee for careful reading of the paper.

Preliminaries
In this section we review the material about embedded contact homology that is needed for the proofs of Theorems 1.2 and 1.5.We include a new, slight extension of the definition of the ECH index to degenerate contact forms.
Throughout this section fix a closed oriented three-manifold Y and a contact form on Y , and let D Ker. / denote the associated contact structure.

Topological preliminaries
We now recall some topological notions we will need, following the treatment in [27].These were originally introduced in a slightly different context in [26].
Definition 2.1 An orbit set is a finite set of pairs ˛D f.˛i ; m i /g where the ˛i are distinct simple Reeb orbits, and the m i are positive integers.We define the homology class of the orbit set ˛by Definition 2.2 If ˛D f.˛i ; m i /g and ˇD f.ˇj ; n j /g are orbit sets with OE˛ D OEˇ, define H 2 .Y; ˛; ˇ/ to be the set of 2-chains Z in Y with @Z D P i m i ˛i P j n j ˇj , modulo boundaries of 3-chains.The set H 2 .Y; ˛; ˇ/ is an affine space over H 2 .Y /.
Given orbit sets ˛and ˇas above, let Z 2 H 2 .Y; ˛; ˇ/, and let be a homotopy class of symplectic trivialization of the contact structure over the Reeb orbits ˛i and ˇj .
where S is a compact oriented surface with boundary, the restriction of f to @S consists of positively oriented covers of f1g ˛i with total multiplicity m i and negatively oriented covers of f 1g ˇj with total multiplicity n j , the composition of f with the projection OE 1; 1 Y !Y represents the class Z, the restriction of f to the interior of S is an embedding, and f is transverse to as follows.Let S and S 0 be admissible representatives of Z and Z 0 , respectively, whose interiors are transverse and do not intersect near the boundary.Define Here # denotes the signed count of intersections, while the remaining terms are linking numbers defined as follows.For > 0 small, the intersection of S with f1 g Y consists of the union over i of a braid C i in a neighborhood of ˛i (see Section 3.1), while the intersection of S with f 1 C g Y consists of the union over j of a braid j in a neighborhood of ˇj .Likewise, S 0 determines braids C i 0 and j 0 .The notation ` indicates the linking number in a neighborhood of ˛i or ˇj computed using the trivialization ; see [27,Section 2.6] for details and sign conventions.
As explained in [27], the relative first Chern class c .˛; ˇ; Z/ and the relative selfintersection number Q .˛;ˇ; Z/ depend only on ˛, ˇ, Z and .Moreover, if we change If is a third orbit set, if is a trivialization of over the Reeb orbits in ˛, ˇand , and if W 2 H 2 .Y; ˇ; /, then we have the additivity properties Note also that the definition of c makes sense more generally if the ˛i and ˇj are transverse knots.Likewise the definition of Q makes sense if the ˛i and ˇj are knots and is an oriented trivialization of their normal bundles.

The ECH index
Let W R=T Z !Y be a Reeb orbit and let be a symplectic trivialization of .The derivative of the time t Reeb flow from .0/ to .t/ , with respect to , is a 2 2 symplectic matrix ˆ.t /.The family of symplectic matrices fˆ.t /g t 2OE0;T induces a family of diffeomorphisms of S 1 in the universal cover of Diff.S 1 /, which has a dynamical rotation number, here denoted by Â ./ 2 R. We call this real number the rotation number of with respect to and denote it by Â ./ 2 R; it depends only on and the homotopy class of .When Â ./ … 1 2 Z, the eigenvalues of the linearized return map (1-2) are e ˙2 iÂ ./ .Definition 2.7 Define the Conley-Zehnder index Remark 2.8 The above definition agrees with the usual Conley-Zehnder index when is nondegenerate.When is degenerate, it is common to give a different definition of the Conley-Zehnder index, as the minimum of the Conley-Zehnder indices of nondegenerate perturbations of , and this will sometimes differ from our definition by 1.For our purposes, especially to obtain an estimate as in Proposition 3.1 below (possibly with a different constant), it does not matter which of these definitions of the Conley-Zehnder index we use for degenerate Reeb orbits.Notation 2.9 If ˛D f.˛i ; m i /g is an orbit set and if is a trivialization of over all of the Reeb orbits ˛i , define CZ .˛ki /: Here k denotes the k th iterate of .It is shown in [31] that @ 2 J D 0. The embedded contact homology ECH .Y; ; ; J / is defined to be the homology of the chain complex .ECC .Y; ; /; @ J /.A theorem of Taubes [43], tensored with Z=2, asserts that there is a canonical isomorphism where the right-hand side is a version of Seiberg-Witten Floer cohomology as defined by Kronheimer and Mrowka [38], and s is a spin-c structure on Y determined by .In particular, ECH depends only 5

Spectral invariants
If ˛D f.˛i ; m i /g is an orbit set, define its symplectic action by Note here that R ˛i agrees with the period of ˛i , because .R/ D 1.
Assume now that is nondegenerate.For L 2 R, define ECC L .Y; ; / to be the subspace of ECC .Y; ; / spanned by ECH generators ˛with symplectic action A.˛/ < L. It follows from the definition of " -compatible almost complex structure" that @ J maps ECC L to itself; see [30,

The ECH index and perturbations
The goal of this section is to prove Proposition 3.1 below, which gives an upper bound on how much the ECH index can change when one perturbs the contact form.This is an important ingredient in the proof of Theorems 1.2 and 1.5.
To state the proposition, let be a contact form on a closed three-manifold Y , and let n D f n be a sequence of contact forms with f n ! 1 in C 2 .In the case of interest, will be degenerate, while each of the contact forms n will be nondegenerate.
Fix an orbit set ˛D f.˛i ; m i /g for , and let N be a disjoint union of tubular neighborhoods N i of the simple Reeb orbits ˛i .Consider a sequence of orbit sets ˛.n/ for n that converges to ˛as currents.In particular this implies that if n is sufficiently large, and if we write ˛0 D ˛.n/, then ˛0 is contained in N , and its intersection with N i is homologous in N i to m i ˛i .There is then a unique W ˛2 H 2 .Y; ˛0; ˛/ that is contained in N .
Likewise fix an orbit set ˇD f.ˇj ; n j /g for along with disjoint tubular neighborhoods of the simple Reeb orbits ˇj , and consider a sequence of orbit sets ˇ.n/ for n that converges to ˇas currents.
Here I.˛; ˇ; Z/ denotes the ECH index for , and I.˛0; ˇ0; Z 0 / denotes the ECH index for n .

Reduction to a local statement
We now reduce Proposition 3.1 to a local statement, Proposition 3.3, below.
Let be an oriented knot in Y , and let N be a tubular neighborhood of with an identification N ' S 1 D 2 .By a "braid in N with d strands", we mean an oriented knot in N which is positively transverse to the D 2 fibers and which intersects each fiber d times.

Definition 3.2
A weighted braid in N with m strands is a finite set of pairs D f. i ; m i /g where the i are disjoint braids in N with d i strands, the m i are positive integers and If is an oriented trivialization of the normal bundle of , then for i ¤ j there is a well-defined linking number ` .i ; j / 2 Z, as discussed in Section 2.1.Similarly, for each i there is a well-defined writhe w .i / 2 Z; see [27,Section 2.6].Define the writhe of the weighted braid by Suppose now that is a simple Reeb orbit for , and that the normal bundle identification N ' S 1 D 2 above is chosen so that the Reeb vector field for is transverse to the D 2 fibers.If 0 D f with f sufficiently C 2 close to 1, then the Reeb vector field for 0 in N is also transverse to the D 2 fibers.Suppose that this is the case.
Let 0 D f. 0 k ; m k /g be an orbit set for 0 which is contained in N .We can regard 0 as a weighted braid with m strands for some positive integer m.Also note that a trivialization of extends to a trivialization of over the entire tubular neighborhood N , and thus canonically induces a homotopy class of trivialization 0 of over the Reeb orbits 0 k .We can now state: Proposition 3.3 With the notation as above, if 0 is sufficiently C 2 close to and if 0 is sufficiently close to m as a current, then Proof of Proposition 3.1 assuming Proposition 3.3 By shrinking the tubular neighborhoods, we can assume without loss of generality that the chosen tubular neighborhood of each orbit ˛i or ˇj has an identification with S 1 D 2 , in which the Reeb flow of is transverse to the D 2 fibers.
In the orbit set ˛0, each pair .˛i; m i / gets replaced by an orbit set ˛0 i which represents a weighted braid with m i strands in the tubular neighborhood of ˛i .Likewise, each pair .ˇj; n j / gets replaced by an orbit set ˇ0 j which represents a weighted braid with n j strands in the tubular neighborhood of ˇj .Let be a homotopy class of symplectic trivializations of over the Reeb orbits ˛i and ˇj .As in Proposition 3.3, this canonically induces a homotopy class of symplectic trivializations 0 over the Reeb orbits in the orbit sets ˛0 i and ˇ0 j .
Because and 0 extend to a trivialization of over the tubular neighborhoods containing W ˛and W ˇ, it follows from the definition of the relative first Chern class that (3-3) c 0 .˛0;ˇ0; Z 0 / D c .˛; ˇ; Z/: CZ .ˇlj / ˇÄ 2n j : By (2-6), (3-3) and (3-4), to complete the proof of Proposition 3.1 it is enough to show To prove (3-5), by Remark 2.6 it is enough to show Since the chosen tubular neighborhoods of the Reeb orbits of ˛i are disjoint, and the chosen tubular neighborhoods of the Reeb orbits of ˇj are disjoint, the above equations follow from Lemma 3.4.
Lemma 3.4 Let D f. i ; m i /g be a weighted braid with m strands as in Definition 3.2.
Let W be the unique relative homology class in H 2 .N; ; .; m//.Then (3-6) Q .; .; m/; W / D w ./: Here defines a trivialization of the vertical tangent bundle of N !which then induces a trivialization of the normal bundle of each braid i .
Proof We can make an admissible representative S for W -see Definition 2.4whose intersection with f1 g N consists of m i parallel (with respect to ) copies of each i , and which shrinks radially towards as the OE 1; 1 coordinate on OE 1; 1 N goes down to 1.We can make another such admissible representative S 0 , disjoint from S , whose intersection with f1 g N is parallel to the first and which likewise shrinks radially towards .Then in (2-1), the intersection number term vanishes.The first linking number term in (2-1) also vanishes, as it is a sum of linking numbers of braids in neighborhoods of the i ; for each i , the braid from S and the braid from S 0 , with respect to , are trivial and parallel, and thus have linking number zero.The second linking number term in (2-1) is a linking number in a neighborhood of and equals w ./.

The structure of the braids
To prove Proposition 3.3, let be a simple Reeb orbit of , let N be a tubular neighborhood of as in Definition 3.2, and let be a trivialization of .Let Â denote the rotation number Â ./ 2 R.
Suppose first that Â is irrational.Then the Reeb orbit and all of its covers are nondegenerate.Consequently, when 0 is sufficiently C 2 close to , there is a unique Reeb orbit 0 0 for 0 close (as a current) to , and for n large the only possibility for the orbit set 0 is that it is the singleton set 0 D f. 0 0 ; m/g.In this case Proposition 3.3 holds because w . 0/ D 0 and the left-hand side of (3-2) is zero.
The nontrivial case of Proposition 3.3 is when the rotation number Â is rational.In this case we need to investigate the braids that can arise in 0 .The idea in what follows is to first analyze the case where the rotation number is an integer, and then reduce the general case to this one by taking an appropriate cover of a neighborhood of .
We start with the case where the rotation number is an integer.Here the picture is simple: each braid has just one strand, and the linking number of any two braids is given by the rotation number.More precisely: Lemma 3.5 With the above notation, suppose that the rotation number Â is an integer a.
(a) For a fixed positive integer d , if f˛ng is a sequence where each ˛n is a simple Reeb orbit for n in N which is a braid with d strands, with ˛n converging as currents to d as n ! 1, then d D 1, and in particular the writhe w .˛n/equals 0 for n large enough.
(b) Given two sequences of simple Reeb orbits f˛ng and fˇng as in (a) with ˛n ¤ ˇn for each n, if n is sufficiently large, then the linking number ` .˛n;ˇn/ equals a.
This lemma is proved in Section 3.3 below.We now consider the case where the rotation number is a rational number a=b that is not an integer.Here there is a similarly nice picture: each new simple Reeb orbit that can appear can be treated, for our purposes, like an .a;b/ torus braid; see also Remark

Now let z
N denote the b-fold cyclic cover of the tubular neighborhood N , with the pullback of the contact form n .There is a unique simple Reeb orbit z 0 0 in z N whose projection to N is a b-fold cover of 0 0 .In addition, by lifting the Reeb orbit ˛n to a Reeb trajectory in z N and extending it by the Reeb flow if needed, we obtain a simple Reeb orbit Q ˛n in z N whose projection to N is a cover of ˛n.By Lemma 3.5(a), if n is sufficiently large, then Q ˛n is a braid with one strand in z N , hence ˛n has at most b strands.Thus, d D b.By Lemma 3.5(b) we have ` .z 0 0 ; Q ˛n/ D a in z N , and it follows that ` . 00 ; ˛n/ D a.We now compute the writhe w .˛n/.There are b possibilities for the Reeb orbit Q ˛n in the previous paragraph, which we denote by Á l for l 2 Z=b, ordered so that the Z=b action on z N by deck transformations sends Á l to Á lC1 .The writhe w .˛n/ is a signed count of crossings of two strands of ˛n.Each such crossing corresponds to a crossing of some Á l with some Á l 0 for l ¤ l 0 , as well as crossings of Á lCp with Á l 0 Cp for p D 1; : : : ; b 1, obtained from the first crossing by deck transformations.On the other hand, the linking number of Á l with Á l 0 is one half the signed count of crossings of Á l with Á l 0 .Thus we obtain Here we are using Lemma 3.5(b) to get that ` .Á l ; Á l 0 / D a when l ¤ l 0 .
We now prove (c).Similarly to the previous calculation, each crossing counted by the linking number ` .˛n;ˇn/ corresponds to b crossings of some lift of ˛n (extended to a simple Reeb obit) with some lift of ˇn (extended to a simple Reeb orbit).Thus the linking number we want is 1=b times the sum of the linking number of each of the b extended lifts of ˛n with each of the b extended lifts of ˇn, which is .1=b/b 2 a D ab.

Perturbations of degenerate flows
To conclude the proof of Proposition 3.3 we now prove Lemma 3.5.
As in the statement of the lemma, let be a simple Reeb orbit of of period T , and let n D f n , where f n ! 1 in C 2 .Let t and t n denote the time t flows of the Reeb vector fields for and n , respectively.Let p 2 , and let P W p ! p denote the linearized return map (1-2).Lemma 3.8 Let f.p n ; T n /g nD1;::: be a sequence in Y .0;1/ satisfying: T n =j n .pn / ¤ p n for all integers j 2 and all n.
Then one of the following alternatives holds: (a2) T 1 D T d for some integer d 2, and the eigenvalues of P that are roots of unity of degree d generate multiplicatively all roots of unity of order d .
Proof This is a special case of a result of Bangert [3, Proposition 1] for C 1 flows.
In the situation of Lemma 3.5, more can be said: Corollary 3.9 Suppose that the eigenvalues of P are real and positive.Let f.p n ; T n /g be a sequence satisfying conditions (c1), (c2) and (c3) of Lemma 3.8.Then alternative (a2) does not hold.
Proof The only root of unity that can be an eigenvalue of P is 1, hence the set of eigenvalues of P does not generate multiplicatively the group of roots of unity of order d when d 2.
Proof of Lemma 3.5 Part (a) follows from Corollary 3.9.
To prove part (b), fix a diffeomorphism ˆfrom the tubular neighborhood N of to .R=T Z/ C such that corresponds to .R=T Z/ f0g, the Reeb vector field R n of n is transverse to the C fibers for n sufficiently large (assume that n is this large below), and the derivative of ˆin the normal direction along agrees with the trivialization .We omit the diffeomorphism ˆfrom the notation below and write points in N using the coordinates .The rotation number of the linearized Reeb flow along differs from the rotation number of any individual vector by less than 1 2 .Hence, by again applying convergence of the n as above, if n is sufficiently large then the rotation number of the path (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) differs by less than 1  2 from a. Since the right-hand side of (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) is an integer which differs by less than 3  4 from a, it must equal a.

Two simple Reeb orbits implies nondegenerate
We now prove Theorem 1.2.Throughout this section assume that Y is a closed connected three-manifold and is a contact form on Y with exactly two simple Reeb orbits 1 and 2 of periods T 1 and T 2 , respectively.Let † be a Seifert surface spanned by i .There is a distinguished homotopy class of trivialization 0 of over i , the "Seifert framing", for which the normal vector to † has winding number zero around i .We have Q i; 0 D 0 by [27, Lemma 3.10].It then follows that i D Â 0 .i /.In the general case when i is rationally nullhomologous, one can similarly describe i as the rotation number with respect to a rational framing of i determined by a rational Seifert surface.
Proof Let l i be a positive integer with l i OE i D 0. Similarly to (4-4), define where c .l i i / is shorthand for c .l i i ; ∅; Z/ for any Z 2 H 2 .Y; l i i ; ∅/.Then c i; does not depend on Z by (2-2) since c 1 ./ is torsion, and it is independent of the choice of l i because c is linear in the relative homology class Z.
It follows from the definition of the ECH index and the facts that c and Q are linear and quadratic in the relative homology class (see [33,Section 4.2]) that where Q i; and Â i; are as in (4)(5).Plugging in the approximation then gives (4-6).To complete the proof of the lemma, it now suffices to show that the sequence .m2;k =m 1;k / k 1 has at least three accumulation points, as then the quadratic form f must vanish identically.We claim that in fact this sequence has infinitely many accumulation points.

Using the volume property
If the sequence has only finitely many accumulation points S 1 ; : : : ; S n , then for every > 0 there exists R > 0 such that every point .m1;k ; m 2;k / is contained in the union of the disk x 2 C y 2 Ä R 2 and the cones around the lines of slope S 1 ; : : : ; S n with angular width .
Since lim k!1 c 2 k =k D 2, and since the points .m1;k ; m 2;k / are pairwise distinct by Proposition 2.18(b), by (4-7) it follows that, for large L, the number of points .m1;k ; m 2;k / contained in the triangle T 1 x C T 2 y Ä L for x 0 and y 0 is approximately 1 2 L 2 .As a result, there exists ı > 0 such that, for all L sufficiently large, the fraction of lattice points in the above triangle that are contained in the sequence .m1;k ; m 2;k / k 1 is at least ı.This gives a contradiction if in the previous paragraph is chosen sufficiently small.

Completing the proof of nondegeneracy
Proof of Theorem 1.2 The ratio T 1 =T 2 is irrational7 by [11,Theorem 1.3].Also, `. 1 ; 2 / is rational by the definition (4-3).It then follows from Lemma 4.6 that 1 and 2 are irrational.By (4-5), since Q i; is rational, it follows that the rotation number Â i; is irrational.Then P i has eigenvalues e ˙2 iÂ i; , so the Reeb orbits i are irrationally elliptic.As explained in Section 1.1, it follows that all covers of i are nondegenerate, so is nondegenerate.

Additional dynamical information
To finish up, we now prove Theorem 1.5.
To prepare for the proof, recall that if Y is a closed oriented three-manifold, if is a contact structure on Y with c 1 ./ D 0 2 H 2 .Y I Z/, and if is a nullhomologous transverse knot, then the self-linking number sl. / 2 Z is defined to be the difference between the Seifert framing (see Remark 4.4) and the framing given by a global trivialization of .In the notation of Section 4.

Definition 2 . 3
(see[27, Section 2.5]) Define the relative first Chern class c .˛; ˇ; Z/ 2 Z as follows.Let S be a compact oriented surface with boundary and let f W S !Y be a smooth map representing the class Z.Let be a section of f which, on each boundary component, is nonvanishing and constant with respect to .Define c .˛; ˇ; Z/ to be the algebraic count of zeroes of .

3 . 7 .Lemma 3 . 6 Remark 3 . 7
More precisely: With the above notation, suppose that the rotation number is Â D a=b, where a and b are relatively prime integers with b > 1.Let n D f n with f n ! 1 in C 2 .Then: (a) For n sufficiently large, there is a unique simple Reeb orbit 0 0 for n that is close to as a current.(b) For a fixed integer d > 1, if f˛ng is a sequence where each ˛n is a simple Reeb orbit for n in N which is a braid with d strands, with ˛n converging as currents to d as n ! 1, then d D b, the writhe w .˛n/equals a.b 1/, and the linking number ` . 00 ; ˛n/ equals a. (c) Given two sequences of Reeb orbits f˛ng and fˇng as in (b) with ˛n ¤ ˇn for each n, if n is sufficiently large, then the linking number ` .˛n;ˇn/ equals ab.In Lemma 3.6(b), we expect that one can further show that if n is sufficiently large then ˛n is an .a;b/ torus braid around 0 0 ; however, we do not need this.Proof of Lemma 3.6 assuming Lemma 3.5 Part (a) holds because the Reeb orbit is nondegenerate.To prove part (b), we first note that, by the same argument as for (a), we must have that d b, because for 0 < d < b the d th iterate of has rotation number da=b … Z.
sl. / D Q ./ c ./;where is any trivialization of j .Now suppose that above is a simple Reeb orbit.Let ./ 2 R denote the rotation number of with respect to the Seifert framing as in Section 4.2, and let Â. / 2 R denote the rotation number of with respect to a global trivialization of .Also, let CZ. / D bÂ. /c C dÂ. /e 2 Z denote the Conley-Zehnder index of with respect to a global trivialization.It follows from (5-1) that (5-2) ./ D Â. / C sl. /: Proof of Theorem 1.5 By Corollary 1.3, 1 and 2 are the core circles of a genus-one Heegaard splitting of Y .It follows from this topological description that `. 1 ; 2 / D 1=p.Part (a) of the theorem then follows from Lemma 4.6.To prove part (b), suppose first that Y D S 3 .We know from Theorem 1.2 that is nondegenerate and there are no hyperbolic Reeb orbits.Then is tight, because otherwise [21, Theorem 1.4] would give a hyperbolic Reeb orbit.Moreover, it follows from [25, Theorem 1.3], combined with [21, Theorem 1.4] and the fact that there are no Reeb orbits with CZ D 2 (since Reeb orbits with even Conley-Zehnder index have integer rotation number and thus are hyperbolic), that one of the simple Reeb orbits, say 1 , satisfies sl. 1 / D 1 and CZ. 1 / D 3, and is the binding of an open book decomposition with pages that are disk-like global surfaces of section for the Reeb flow.The return map on a page preserves an area form with finite total area, and hence it has a fixed point by Brouwer's translation theorem.This fixed point corresponds to the simple Reeb orbit 2 , which is transverse to the pages of the open book.Since, on S 3 n 1 , the tangent spaces of the pages define a distribution that is isotopic to keeping transversality with the Reeb direction, sl. 2 / D 1. Since CZ. 1 / D 3, we have Â. 1 / 2 .1;2/, so, by (5-2), 1 2 .0;1/.By Lemma 4.6 as used in (1-4), we have 1.10Let Y be a closed three-manifold , and let be a contact form on Y with exactly two simple Reeb orbits.Then both orbits bound disk-like global surfaces of section whose associated return maps define smooth pseudorotations of the open disk.
Definition 2.10 Let ˛and ˇbe orbit sets with OE˛ D OEˇ 2 H 1 .Y /, and Z 2 H 2 .Y; ˛; ˇ/.Cristofaro-Gardiner, Umberto Hryniewicz, Michael Hutchings and Hui LiuRemark 2.13 In the special case where c 1 ./ 2 H 2 .Y I Z/ is torsion and D 0, the chain complex ECC .Y; ; 0/ has a canonical absolute Z-grading defined by Definition 2.14 An almost complex structure J on R Y is -compatible if J @ s D R, where s denotes the R coordinate, J is invariant under the R action on R Y by translation of s, and J. / D , rotating positively with respect to d .
[11,3)(4)(5)(6)(7)(8)(9)(10)(11) J .However, unlike the usual ECH, filtered ECH does depend on the contact form and not just on the contact structure .Cristofaro-Gardiner, Umberto Hryniewicz, Michael Hutchings and Hui LiuWe can change the contact form , without changing the contact structure , by multiplying by a smooth function f W Y !R >0 .As explained in [11, Section 2.5], it turns out that even when is degenerate, one can still define c .Y; / as a limit of spectral invariants c .Y; f n / where f n is nondegenerate and f n ! 1 in C 0 .Proposition 2.18 Let Y be a closed connected three-manifold , and let be a (possibly degenerate) contact form on Y .Then:If is nondegenerate, then since the chain map U J;z counts J -holomorphic curves it decreases symplectic action like the differential, so strict inequality in (2-11) holds.The not necessarily strict inequality(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)in the degenerate case follows by a limiting argument.The fact that (2-11) is strict for degenerate contact forms with only finitely many simple Reeb orbits 6 is proved by a more subtle compactness argument for holomorphic curves in[11, Lemma 3.1(b)].
[34,ion 1.4].We define the filtered ECH to be the homology of this subcomplex, which we denote by ECH L .Y; ; /.The inclusion of chain complexes induces a map{ L W ECH L .Y; ; / !ECH .Y; ; /:It is shown in[34, Theorem 1.3] that the filtered homology ECH L .Y; ; / and the map { Definition 2.17 [29] If 0 ¤ 2 ECH .Y; ; /, define the spectral invariant c .Y; / D inffL j 2 Im.{ L /g 2 R:An equivalent definition is that c .Y; / is the minimum L such that the class can be represented by a cycle in the chain complex .ECC .Y; ; /; @ J / which is a sum of ECH generators each having symplectic action Ä L. In particular, by definition, c .Y; / D A.˛/ for some ECH generator ˛with OE˛ D .Daniel (a) If 0 ¤ 2 ECH .Y; ; /, then c .Y; / D A.˛/ for some orbit set ˛with OE˛ D .(b) If 2 ECH .Y; ; / and U ¤ 0, then (2-11) c U .Y; / Ä c .Y; /: If there are only finitely many simple Reeb orbits, then the inequality (2-11) is strict.(c) Volume property If c 1 ./ C 2 PD./ 2 H 2 .Y I Z/ is torsion, and if f k g k 1 is a U -sequence for , then lim k!1 c k .Y; / 2 k D 2 vol.Y; /: Proof As noted above, part (a) holds by definition when is nondegenerate, and in the degenerate case it follows from a compactness argument for Reeb orbits; see [11, Lemma 3.1(a)].
t; z/ 2 .R=T Z/ C. By part (a), by taking n large enough we can assume that ˛n and ˇn have the same period as .After reparametrization, the Reeb orbit ˛n is given by a map R=T Z !.R=T Z/ C; t 7 !.t;O ˛n.t //; where O ˛n W R=T Z ! C. Likewise the Reeb orbit ˇn is given by a map O ˇn W R=T Z ! C. !C such that, for z 2 U , the flow of the Reeb vector field R n starting at .0; z/ first hits ftg C at the point .t; [12,classes OE i 2 H 1 .Y / and c 1 ./ 2 H 2 .Y I Z/ are torsion.Proof We use a similar argument to the proof of[12, Theorem 1.7].Since every oriented three-manifold is spin, we can choose 2 H 1 .Y / such that c 1 ./C 2 PD./ D 0 2 H 2 .Y I Z/.By Proposition 2.16, there exists a U -sequencef k g 1 for .Write c k D c k .Y; / 2 R. OE 1 C m 2;k OE 2 D 2 H 1 .Y /:By Proposition 2.18(b), the sequence fc k g is strictly increasing.It then follows from (4-1) that there are infinitely many integral linear combinations of OE 1 and OE 2 that have the same value in H 1 .Y /.Thus the kernel of the map(4-2) Z 2 !H 1 .Y /; .m 1 ; m 2 / 7 !m 1 OE 1 C m 2 OE 2 ;In fact, the kernel of the map (4-2) must have rank at least 2; otherwise c k would grow at least linearly in k, contradicting the sublinear growth in the volume property in Proposition 2.18(c).It follows that OE 1 and OE 2 are torsion.Since c 1 ./C2PD./D 0, we deduce that c 1 ./ is also torsion.4.2 Computing the ECH indexIf m 1 and m 2 are nonnegative integers, we use the notation m 1 ; m 1 /; . 2 ; m 2 /g, with the element .i ; m i / omitted when m i D 0. Write ˛D m 1 .If OE˛ D 0, then it follows from Remark 2.13 and Lemma 4.1 that I.˛/ 2 Z is defined.We now give an explicit computation of I.˛/, following [33, Section 4.7].
1 2where l 1 and l 2 are positive integers such that l i OE i D 0 2 H 1 .Y /, and on the right-hand side `denotes the usual integer-valued linking number of disjoint nullhomologous loops.Definition 4.3 For i D 1; 2, define the Seifert rotation number i 2 R as follows.Let be a trivialization of over i .Let Â i; D Â .i / 2 R denote the rotation number of i with respect to .Let l i be a positive integer such that l i OE i D 0. Define (4-4) Q for any Z 2 H 2 .Y; l i i ; ∅/.Note that Q i; does not depend on Z by (2-3), and it does not depend on l i either because Q is quadratic in the relative homology class.Finally, define Remark 4.4 When i is nullhomologous, one can alternatively describe i as follows.
i WD Q i; C Â i; 2 R:The number i does not depend on the choice of trivialization by the change of trivialization formulas in [27,Section 2].
By Proposition 2.16 and Lemma 4.1, we can choose a U -sequence f k g k 1 for D 0. Since the U map has degree 2, there is a constant C 2 Z such that, for each positive integer k, the class k has grading C C 2k.By Proposition 2.18(a), for each positive integer k there are nonnegative integers m 1;k and m 2;k such that(4-7) c k .Y; / D m 1;k T 1 C m 2;k T 2 :.If 0 is a sufficiently C 2 close nondegenerate perturbation of , then by the same compactness argument that proves Proposition 2.18(a) there is an orbit set ˛0 k close to ˛k as a current such that I.˛0 k / D C C 2k (and also R ˛0 k 0 is close to c k .Y; /, although we do not need this).By Proposition 3.1,C C 2k D I.˛k/ C O.m 1;k C m 2;k /:Consequently, if the sequence .m2;k =m 1;k / k 1 has an accumulation point S 2 OE0; 1, then the line in the .x;y/-plane of slope S through the origin is in the null space of the quadratic form f .x;y/ D . 1 T 2 1 /x 2 C . 2 T 2 2 /y 2 C 2.`. 1 ; 2 / T 1 T 2 /xy: Proof Both sides of the above equations are invariant under scaling the contact form by a positive constant, so we may assume without loss of generality that vol.Y;/ D 1. 1 ; 2 /m 1;k m 2;k C O.m 1;k C m 2;k /: 1 ; 2 / T 1 T 2 /m 1;k m 2;k D O.m 1;k C m 2;k / C o.k/: