# Proof of Hofer-Wysocki-Zehnder’s two or infinity conjecture

Dan Cristofaro-Gardiner  ,  Umberto Hryniewicz  ,  Michael Hutchings  and  Hui
Liu

###### Abstract.

We prove that every Reeb flow on a closed connected three-manifold has either
two or infinitely many simple periodic orbits, assuming that the associated
contact structure has torsion first Chern class. As a special case, we prove a
conjecture of Hofer-Wysocki-Zehnder published in 2003 asserting that a smooth
and autonomous Hamiltonian flow on $\mathbb{R}^{4}$ has either two or
infinitely many simple periodic orbits on any regular compact connected energy
level that is transverse to the radial vector field. Other corollaries settle
some old problems about Finsler metrics: we show that every Finsler metric on
$S^{2}$ has either two or infinitely many prime closed geodesics; and we show
that a Finsler metric on $S^{2}$ with at least one closed geodesic that is not
irrationally elliptic must have infinitely many prime closed geodesics. The
novelty of our work is that we do not make any nondegeneracy hypotheses.

###### Contents

  1. 1 Introduction
    1. 1.1 The main result and corollaries
    2. 1.2 Some history and related results
    3. 1.3 Idea of the proof and organization of the paper
    4. 1.4 Some open questions and a small step
  2. 2 Preliminaries on embedded contact homology
    1. 2.1 Orbit sets and the ECH index
    2. 2.2 Holomorphic currents and the index inequality
    3. 2.3 The definition of ECH
    4. 2.4 The $U$ map
    5. 2.5 Spectral invariants and the volume property
    6. 2.6 Partition conditions
    7. 2.7 The $J_{0}$ index and topological complexity bounds
    8. 2.8 Criterion for a Birkhoff section
  3. 3 Birkhoff sections for nondegenerate perturbations
    1. 3.1 Introduction
    2. 3.2 Setting the stage
      1. 3.2.1 The perturbation lemma
      2. 3.2.2 Setting up the proof of Proposition 3.1
      3. 3.2.3 Basic estimates
      4. 3.2.4 Exceptional pairs
      5. 3.2.5 Some constants
      6. 3.2.6 More basic estimates
      7. 3.2.7 A chain of $U$-curves
    3. 3.3 The score
      1. 3.3.1 Definition of the score
      2. 3.3.2 How orbits contribute to the score
      3. 3.3.3 Nonnegativity of the score
    4. 3.4 Putting it all together
      1. 3.4.1 Finding a low energy cylinder
      2. 3.4.2 Conclusion
  4. 4 Preliminaries about holomorphic curves in symplectizations
    1. 4.1 Generalities
    2. 4.2 Weighted Fredholm theory and automatic transversality
  5. 5 Asymptotic analysis and compactness
    1. 5.1 Localizing the ends
    2. 5.2 Special coordinates
    3. 5.3 Uniform exponential decay
    4. 5.4 Asymptotic data of the cylinder ${\widetilde{u}}$
    5. 5.5 Compactness
  6. 6 Two or infinitely many simple Reeb orbits

##  1\. Introduction

###  1.1. The main result and corollaries

Let $Y$ be a closed oriented three-manifold. A contact form on $Y$ is a
$1$-form $\lambda$ such that $\lambda\wedge d\lambda>0$ everywhere. Given a
contact form $\lambda$, the two-plane field
$\xi=\operatorname{Ker}(\lambda)\subset TY$ is oriented by $d\lambda$, and
this oriented two-plane field is called the contact structure associated to
$\lambda$. In addition, $\lambda$ has an associated Reeb vector field $R$
defined by the equations

$d\lambda(R,\cdot)=0,\quad\quad\lambda(R)=1.$

The flow of $R$ is called the Reeb flow. A periodic orbit of the Reeb flow,
which we call a Reeb orbit for short, is a map
$\gamma:\mathbb{R}/T\mathbb{Z}\to Y$ for some $T>0$ such that
$\gamma^{\prime}(t)=R(\gamma(t))$. Here $T$ is the period of $\gamma$. We
declare two Reeb orbits to be equivalent if they differ by reparametrizing the
domain $\mathbb{R}/T\mathbb{Z}$ by a translation. Any Reeb orbit $\gamma$ is a
degree $d$ cover of its image for some $d\geq 1$. We refer to the integer $d$
as the covering multiplicity of $\gamma$, and we say that $\gamma$ is simple
when $d=1$, i.e. the map $\gamma$ is an embedding.

Our main result is the following:

######  Theorem 1.1.

Let $Y$ be a closed connected three-manifold, let $\lambda$ be a contact form
on $Y$, and let $\xi=\operatorname{Ker}(\lambda)$ be the associated contact
structure. Assume that the first Chern class $c_{1}(\xi)\in
H^{2}(Y;\mathbb{Z})$ is torsion. Then $\lambda$ has either two or infinitely
many simple Reeb orbits.

######  Remark 1.2.

The conclusion of Theorem 1.1 was previously shown in [9] under the additional
hypothesis that the contact form $\lambda$ is nondegenerate (see §2.1 for the
definition of “nondegenerate”). Subsequently, the hypothesis on the first
Chern class was removed by Colin-Dehornoy-Rechtman [6], but still assuming
nondegeneracy. The novelty of Theorem 1.1 is that we do not require any
genericity hypothesis on the Reeb flow. See §1.2 for more discussion of
history and related results.

The following is an important example of Theorem 1.1. Let
$Y\subset\mathbb{R}^{4}$ be a compact connected smooth hypersurface which is
transverse to the radial vector field. Following the symplectic geometry
convention, we call such a hypersurface star-shaped. Indeed, $Y$ bounds a
compact domain which is star-shaped in the usual sense. The standard Liouville
form on $\mathbb{R}^{4}$ defined by

(1.1)
$\lambda=\frac{1}{2}\sum_{i=1}^{2}\left(x_{i}\,dy_{i}-y_{i}\,dx_{i}\right)$

restricts to a contact form on $Y$. If $H:\mathbb{R}^{4}\to\mathbb{R}$ is a
smooth function having $Y$ as a regular level set, then the Reeb vector field
of $\lambda|_{Y}$ is parallel to the Hamiltonian vector field $X_{H}$ on $Y$,
so that Reeb orbits are the same as periodic orbits of $X_{H}$ on $Y$ up to
reparametrization.

Since $Y$ is diffeomorphic to $S^{3}$, Theorem 1.1 is applicable to this
example to prove the following result, which was a conjecture of Hofer-
Wysocki-Zehnder from [24]; Hofer-Wysocki-Zehnder proved this in [24], under
the additional hypothesis that the contact form is nondegenerate and the
stable and unstable manifolds of all hyperbolic Reeb orbits intersect
transversely.

######  Corollary 1.3.

[24, Conjecture 1.13] Every star-shaped hypersurface in $\mathbb{R}^{4}$, with
the restriction of the standard Liouville form (1.1), has either two or
infinitely many simple Reeb orbits.

Theorem 1.1 also has implications for Finsler geometry:

######  Corollary 1.4.

[1, Problem 15] [4, Conjecture 2.2.1] [41, Conjecture 1] Every Finsler metric
on $S^{2}$ has either two or infinitely many prime closed geodesics.

###### Proof.

Let $F$ be a Finsler metric on $S^{2}$, and let $Y$ be the unit tangent bundle
of $S^{2}$ associated to $F$. As reviewed in [26, §2.3.3], there exists a
contact form $\lambda_{F}$ on $Y$ such that prime closed geodesics on $S^{2}$
are in bijective correspondence with simple Reeb orbits on $Y$ under the
projection map $Y\to S^{2}$. Thus the corollary follows from Theorem 1.1,
since $H^{2}(Y;\mathbb{Z})$ is torsion. ∎

######  Remark 1.5.

It has been a longstanding conjecture ([1, Problem 15], [4, Conjecture 2.2.1],
[41, Conjecture 1]) that every irreversible Finsler metric on $S^{2}$ has
either two or infinitely many prime closed geodesics. Corollary 1.4 settles
this conjecture. To put this result in context, it has long been known that
there are infinitely many prime closed geodesics on every closed surface with
a Riemannian metric; for positive genus surfaces this is due to Hadamard [17],
and the genus zero case was proved by Bangert and Franks [3, 13]. The results
in the Riemannian case remain true in the Finsler case for positive genus
surfaces, and can be proved by similar methods. However, the genus zero case
presents a strong contrast, as Katok [39] has constructed Finsler metrics on
$S^{2}$ with exactly two prime closed geodesics.

######  Remark 1.6.

A theorem outlined by Lyusternik-Schnirelmann [43] and proved by Grayson [16,
Corollary 0.2] asserts that every Riemannian metric on $S^{2}$ has at least
three distinct embedded closed geodesics. Hence Corollary 1.4 recovers the
result of Bangert and Franks [3, 13] asserting that every Riemannian metric on
$S^{2}$ has infinitely many prime closed geodesics.

Our previous work [7] established strong restrictions in the case of two
simple Reeb orbits, and so we also obtain as a corollary various criteria
guaranteeing the existence of infinitely many simple Reeb orbits:

######  Corollary 1.7.

Let $Y$ be a closed three-manifold and let $\lambda$ be a contact form on $Y$
such that the associated contact structure $\xi=\operatorname{Ker}(\lambda)$
has $c_{1}(\xi)\in H^{2}(Y;\mathbb{Z})$ torsion. Assume that at least one of
the following is true:

  1. (a)

$Y$ is not a lens space.

  2. (b)

There is at least one Reeb orbit which is not irrationally elliptic (see §2.1
for the definition of “irrationally elliptic”).

Then there are infinitely many simple Reeb orbits.

###### Proof.

Without loss of generality, $Y$ is connected. By [7, Thm. 1.2, Cor. 1.3], if
there are exactly two simple Reeb orbits, then both simple Reeb orbits (and
all their iterates) are irrationally elliptic, and $Y$ is a lens space,
contrary to our hypothesis. Hence by the dichotomy in Theorem 1.1, there are
infinitely many simple Reeb orbits. ∎

The above corollary in turn implies the following result, which is slightly
stronger than a conjecture of Long [4, Conjecture 2.2.2] about the kinds of
geodesics that must appear on a Finsler $S^{2}$.

######  Corollary 1.8.

A Finsler metric on $S^{2}$ with a closed geodesic that is not irrationally
elliptic has infinitely many prime closed geodesics.

###### Proof.

Apply Corollary 1.7(b) to the contact form on the unit tangent bundle of
$S^{2}$ determined by the Finsler metric, as in Corollary 1.4. ∎

One could alternatively deduce Corollary 1.8 from Corollary 1.4 as a
consequence of [42, Thm. 1.2].

###  1.2. Some history and related results

The three-dimensional case of the Weinstein conjecture [51] asserts that every
contact form on a closed three-manifold has at least one Reeb orbit. This was
proved for star-shaped hypersurfaces in $\mathbb{R}^{4}$ using variational
methods by Rabinowitz [44], who also proved a higher dimensional analogue in
$\mathbb{R}^{2n}$. The Weinstein Conjecture was proved for all contact forms
on $S^{3}$, and for all overtwisted contact forms on any closed three-
manifold, in work of Hofer [19] where the foundations of holomorphic curve
theory in symplectizations were first introduced and developed. The Weinstein
conjecture was established for all closed three-manifolds by Taubes [48] using
Seiberg-Witten theory; see [31] for a survey. Taubes subsequently generalized
this work to prove an isomorphism between embedded contact homology (ECH) and
Seiberg-Witten Floer cohomology reviewed in §2.3 below. This isomorphism,
together with a “Weyl law” for the ECH spectrum reviewed in §2.5 below, was
used in [8] to show that in fact, every contact form on a closed three-
manifold has at least two simple Reeb orbits. It was proved in [15] that the
symplectic field theory of Eliashberg-Givental-Hofer [12] can be combined with
arguments of Hingston [18] and Ginzburg [14] to prove that the Reeb flow has
at least two simple closed orbits in the special case of star-shaped
hypersurfaces in $\mathbb{R}^{4}$. A detailed description of the dynamics when
there are exactly two simple Reeb orbits was given in [7], using a result from
[36] on the nondegenerate case. Such examples are quite special; among other
things, the three-manifold must be diffeomorphic to $S^{3}$ or a lens space.
In fact it was shown by Irie [38] that for a $C^{\infty}$ generic contact form
on a closed three-manifold $Y$, not only are there infinitely many simple Reeb
orbits, but moreover the union of the images of the Reeb orbits is dense in
$Y$.

The two or infinity dichotomy in Corollary 1.3 was proved for the special case
of convex hypersurfaces by Hofer-Wysocki-Zehnder [23]. The main ingredient in
the proof is to show that in this case there is always a disk-like global
surface of section. This perspective is in fact central to our work, as the
notion of a global surface of section, or more precisely the slightly more
general notion of a Birkhoff section (see §2.8 for the definition), plays a
key role in our arguments. As mentioned above, in [24], Hofer-Wysocki-Zehnder
proved the conclusion of Corollary 1.3 for star-shaped hypersurfaces under the
additional hypothesis that the contact form is nondegenerate and the stable
and unstable manifolds of all hyperbolic Reeb orbits intersect transversely.
This work introduced the theory of finite energy foliations, which generalizes
Birkhoff sections.

As mentioned in Remark 1.2, Colin-Dehornoy-Rechtman [6] proved that there are
two or infinitely many simple Reeb orbits for any nondegenerate contact form
on a closed connected three-manifold. This is a consequence of a general
existence result for “broken book decompositions”. The latter are analogues of
finite-energy foliations, and were used in [6] to establish deep structural
results for Reeb flows of nondegenerate contact forms on closed three-
manifolds.

###  1.3. Idea of the proof and organization of the paper

The strategy of the proof of Theorem 1.1 is as follows. Suppose that there are
only finitely many simple Reeb orbits. Following the strategy in [9], we would
like to use holomorphic curves to find a genus zero Birkhoff section for the
Reeb flow. Roughly speaking, this is an open surface $\Sigma\subset Y$ such
that the simple Reeb orbits correspond to ends of $\Sigma$, together with
simple periodic orbits of an area-preserving return map
$\varphi:\Sigma\to\Sigma$. If such a genus zero Birkhoff section can be found,
then one can apply a theorem of Franks [13] to deduce that there are either
two or infinitely many simple Reeb orbits.

The challenge with this approach is to deal with the case when the contact
form $\lambda$ is degenerate. Much of the theory of holomorphic curves that we
would want to use, including embedded contact homology (ECH), is only defined
for nondegenerate contact forms. To get started, we can find a sequence of
nondegenerate contact forms $\lambda_{k}$ converging to $\lambda$. After
reviewing necessary preliminaries on ECH in §2, in §3 we show that when $k$ is
sufficiently large, $\lambda_{k}$ admits a genus zero Birkhoff section. In
preparation for taking a limit as $k\to\infty$, we need to know more: We want
each such Birkhoff section to be an annulus, with a uniform upper bound on the
periods of the boundary Reeb orbits, and to satisfy additional technical
conditions. This requires lengthy combinatorial arguments involving details of
the holomorphic curves arising from ECH.

It would now be natural to try to show that the annular Birkhoff sections for
$\lambda_{k}$ can be taken to converge to an annular Birkhoff section for
$\lambda$. We do not quite prove this, although it may be true, but we prove a
slightly weaker statement which is sufficient for our purposes. Namely, the
limits of the boundary Reeb orbits of the Birkhoff sections give a two-
component link $\mathcal{L}$ in $Y$ whose components are simple Reeb orbits of
$\lambda$. In addition, there is a moduli space $\mathcal{M}$ consisting of
holomorphic cylinders with one marked point, and a covering map
$\mathcal{M}\to Y\setminus\mathcal{L}$. Finally, the Reeb flow on
$Y\setminus\mathcal{L}$ lifts to a flow on $\mathcal{M}$ to which Franks’s
theorem can be applied to complete the proof of Theorem 1.1. Carrying out this
program requires considerable technical work to deal with holomorphic curves
asymptotic to possibly degenerate Reeb orbits, and this is the topic of
Sections 4, 5, and 6.

###  1.4. Some open questions and a small step

It remains an open question whether the torsion hypothesis can be removed from
Theorem 1.1. By [7, Cor. 1.3], this is equivalent to the question of whether
every contact form on a closed three-manifold that is not a lens space must
have infinitely many simple Reeb orbits.

In the cases where there are countably infinitely many simple Reeb orbits, one
can ask for lower bounds on the growth rate of the number of simple Reeb
orbits with period $\leq T$ as $T\to\infty$. Our methods allow a small step in
this direction. Enumerate the simple Reeb orbits as
$\gamma_{1},\gamma_{2},\ldots$ such that the period of $\gamma_{i}$ is a
nondecreasing function of $i$. Then it follows from Proposition 3.4 below that
for a torsion contact form on a three-manifold other than $S^{3}$ or a lens
space, we obtain an upper bound (possibly very large) on the period of
$\gamma_{n+1}$ in terms of data associated to $\gamma_{1},\ldots,\gamma_{n}$.

### Acknowledgements

Key discussions for this project occurred during the conference “Low-
dimensional topology and homeomorphism groups”, hosted by the Brin Mathematics
Research Center at the University of Maryland. We thank the Brin Center for
their support.

DCG thanks Dan Pomerleano for numerous very helpful discussions related to the
present work; our earlier joint paper [9] also contained many insights which
were crucial for this one. DCG also thanks the NSF for their support under
agreements DMS-2227372 and DMS-2238091.

UH thanks Pedro Salomão for various discussions about obtaining global
surfaces of section for degenerate Reeb flows, which were crucial to the
current paper. UH acknowledges the support by DFG SFB/TRR 191 ‘Symplectic
Structures in Geometry, Algebra and Dynamics’, Projektnummer 281071066-TRR
191.

MH was partially supported by NSF grant DMS-2005437.

HL was partially supported by NSFC (Nos. 12371195, 12022111) and the
Fundamental Research Funds for the Central Universities (No. 2042023kf0207)

##  2\. Preliminaries on embedded contact homology

We now review the background on embedded contact homology (ECH) that we will
need. More details about the basic definitions may be found for example in
[33].

###  2.1. Orbit sets and the ECH index

Let $Y$ be a closed $3$-manifold, and $\lambda$ be a contact form on $Y$. Let
$\xi=\operatorname{Ker}\lambda$ denote the associated contact structure, and
let $R$ denote the Reeb vector field on $Y$ determined by $\lambda$. If
$\gamma:\mathbb{R}/T\mathbb{Z}\to Y$ is a Reeb orbit, then the derivative of
the Reeb flow at time $T$ defines a linearized return map

$P_{\gamma}:\xi_{\gamma(0)}\longrightarrow\xi_{\gamma(0)},$

which is symplectic with respect to $d\lambda$. We say that the Reeb orbit
$\gamma$ is nondegenerate if $1$ is not an eigenvalue of $P_{\gamma}$. We say
that the contact form $\lambda$ is nondegenerate if all Reeb orbits are
nondegenerate. Let $\gamma$ be a Reeb orbit. If both eigenvalues of
$P_{\gamma}$ lie on the unit circle we say that $\gamma$ is elliptic,
otherwise we say that $\gamma$ is hyperbolic. In the hyperbolic case both
eigenvalues of $P_{\gamma}$ are real and lie outside the unit circle, and they
are both positive or both negative; in the former case $\gamma$ is said to be
positive hyperbolic, in the latter case $\gamma$ is said to be negative
hyperbolic. In the elliptic case, we call $\gamma$ irrationally elliptic if
the eigenvalues of $P_{\gamma}$ are of the form $e^{\pm i2\pi\alpha}$ for some
irrational number $\alpha\in\mathbb{R}\setminus\mathbb{Q}$. When $\gamma$ and
all its iterates are nondegenerate then there are three mutually exclusive
cases: positive hyperbolic, negative hyperbolic or irrationally elliptic.

If $\tau$ is a $d\lambda$-symplectic trivialization of
$\gamma^{*}\xi\to\mathbb{R}/T\mathbb{Z}$, then the linearized Reeb flow along
$\gamma$ in the trivialization $\tau$ has a well-defined rotation number
$\theta\in\mathbb{R}$. If $\gamma$ is nondegenerate, then the rotation number
$\theta$ is an integer when $\gamma$ is positive hyperbolic, $1/2$ plus an
integer when $\theta$ is negative hyperbolic, and irrational when $\gamma$ is
elliptic and all iterates of $\gamma$ are nondegenerate. If $\gamma$ is
nondegenerate111If $\gamma$ is degenerate, there are different conventions for
the Conley-Zehnder index. It would be possible to use (2.1) to define the
Conley-Zehnder index in the degenerate case and we used this convention in
[7]. However in §4 we will use a different convention which in the degenerate
case is sometimes one less than (2.1)., the Conley-Zehnder index of $\gamma$
with respect to $\tau$ is given by

(2.1) ${\rm CZ}_{\tau}(\gamma)=\lfloor\theta\rfloor+\lceil\theta\rceil.$

For the rest of the discussion of ECH we assume unless otherwise stated that
$\lambda$ is nondegenerate; this holds for $C^{\infty}$-generic contact forms.

######  Definition 2.1.

An orbit set is a finite set of pairs $\alpha=\\{(\alpha_{i},m_{i})\\}$ where
the $\alpha_{i}$ are distinct simple Reeb orbits and the $m_{i}$ are positive
integers. The homology class of the orbit set $\alpha$ is defined to be

$[\alpha]=\sum_{i}m_{i}[\alpha_{i}]\in H_{1}(Y)$

and its symplectic action of $\alpha$ is defined by

$\mathcal{A}(\alpha)=\sum_{i}m_{i}\int_{\alpha_{i}}\lambda.$

Note here that $\int_{\alpha_{i}}\lambda$ is simply the period of
$\alpha_{i}$. An ECH generator is an orbit set
$\alpha=\\{(\alpha_{i},m_{i})\\}$ such that $m_{i}=1$ whenever $\alpha_{i}$ is
hyperbolic.

If $\alpha=\\{(\alpha_{i},m_{i})\\}$ and $\beta=\\{(\beta_{j},n_{j})\\}$ are
orbit sets with $[\alpha]=[\beta]\in H_{1}(Y)$, define $H_{2}(Y,\alpha,\beta)$
to be the set of relative homology classes of $2$-chains $Z$ in $Y$ with

$\partial Z=\sum_{i}m_{i}\alpha_{i}-\sum_{j}n_{j}\beta_{j},$

modulo boundaries of $3$-chains. The set $H_{2}(Y,\alpha,\beta)$ is an affine
space over $H_{2}(Y)$.

######  Definition 2.2.

Given $Z\in H_{2}(Y,\alpha,\beta)$ as above, the ECH index is defined by

$I(\alpha,\beta,Z)=c_{\tau}(Z)+Q_{\tau}(Z)+\sum_{i}\sum_{k=1}^{m_{i}}{\rm
CZ}_{\tau}(\alpha_{i}^{k})-\sum_{j}\sum_{k=1}^{n_{j}}{\rm
CZ}_{\tau}(\beta_{j}^{k})\in\mathbb{Z}.$

Here $\tau$ is a trivialization of the contact structure $\xi$ over the Reeb
orbits $\alpha_{i}$ and $\beta_{i}$; the ECH index does not depend on the
choice of $\tau$, although the individual terms in it do. The term
$c_{\tau}(Z)$ is the relative first Chern class $c_{1}(f^{*}\xi,\tau)$, where
$f:\Sigma\to Y$ is a smooth map from a compact oriented surface $\Sigma$ with
boundary mapping to $\alpha-\beta$, representing the relative homology class
$Z$. This relative first Chern class is defined by choosing a generic section
of $f^{*}\xi$ over $\Sigma$ which on each boundary component is nonvanishing
and has winding number zero with respect to $\tau$, and then counting its
zeroes with signs. The term $Q_{\tau}(Z)$ is the relative intersection
pairing; the detailed definition is not needed here and may be found in [33,
§3.3] or [30, §2.7].

For orbit sets $\alpha$ and $\beta$ with $[\alpha]=[\beta]=\Gamma\in
H_{1}(Y)$, and for a trivialization $\tau$ over the Reeb orbits in $\alpha$
and $\beta$, the above notions depend on the relative homology class $Z$ as
follows: Given another relative homology class $W\in H_{2}(Y,\alpha,\beta)$,
one can form the difference $Z-W\in H_{2}(Y)$, and we have

(2.2) $\displaystyle c_{\tau}(Z)-c_{\tau}(W)$ $\displaystyle=\langle
Z-W,c_{1}(\xi)\rangle,$ $\displaystyle Q_{\tau}(Z)-Q_{\tau}(W)$
$\displaystyle=\langle Z-W,2\operatorname{PD}(\Gamma)\rangle.$

Here $c_{1}(\xi)\in H^{2}(Y;\mathbb{Z})$ denotes the usual first Chern class
of $\xi$, and $\operatorname{PD}(\Gamma)\in H^{2}(Y;\mathbb{Z})$ is the
Poincaré dual of $\Gamma$. It follows that we have the “index ambiguity
formula”

(2.3) $I(\alpha,\beta,Z)-I(\alpha,\beta,W)=\langle
Z-W,c_{1}(\xi)+2\operatorname{PD}(\Gamma)\rangle.$

###  2.2. Holomorphic currents and the index inequality

######  Definition 2.3.

An almost complex structure $J$ on $\mathbb{R}\times Y$ is
$\lambda$-compatible if:

  * •

$J(\partial_{a})=R$, where $a$ denotes the $\mathbb{R}$ coordinate.

  * •

$J$ is invariant under translation of the $\mathbb{R}$ factor.

  * •

$J$ sends the contact structure $\xi$ to itself, rotating positively with
respect to $d\lambda$.

Given a $\lambda$-compatible almost complex structure $J$, we can consider a
$J$-holomorphic map

$u:(\Sigma,j)\longrightarrow(\mathbb{R}\times Y,J)$

where the domain $\Sigma$ is a compact connected Riemann surface without
boundary, with finitely many punctures, and for each puncture there is a
neighborhood which maps asymptotically to a Reeb orbit with $a\to\pm\infty$;
for more precise definitions, see §4. We declare two such holomorphic maps to
be equivalent if they agree up to biholomorphic reparametrization of the
domain. Equivalence classes are called $J$-holomorphic curves. If the map $u$
is somewhere injective, then we can identify the associated curve with its
image in $\mathbb{R}\times Y$, which we typically denote using the letter $C$.

######  Example 2.4.

If $\gamma$ is any simple Reeb orbit, then the “trivial cylinder”
$\mathbb{R}\times\gamma\subset\mathbb{R}\times Y$ is $J$-holomorphic, by the
definition of “$\lambda$-compatible”.

We define a $J$-holomorphic current to be a finite linear combination

$\mathcal{C}=\sum_{i}d_{i}C_{i}$

where the $C_{i}$ are distinct somewhere injective $J$-holomorphic curves as
above, and the $d_{i}$ are positive integers. Note that we use the
calligraphic letter $\mathcal{C}$ to denote a current, and the ordinary letter
$C$ to denote a somewhere injective curve.

For a $J$-holomorphic current $\mathcal{C}$ as above, there are orbit sets
$\alpha$ and $\beta$ such that as currents, we have
$\lim_{a\to\infty}\mathcal{C}\cap(\\{a\\}\times Y)=\alpha$ and
$\lim_{a\to-\infty}\mathcal{C}\cap(\\{a\\}\times Y)=\beta$. We call $\alpha$
the positive orbit set of $\mathcal{C}$, and we call $\beta$ the negative
orbit set of $\mathcal{C}$. The projection of the current $\mathcal{C}$ to $Y$
defines a relative homology class

$[\mathcal{C}]\in H_{2}(Y,\alpha,\beta).$

We define the ECH index of $\mathcal{C}$ by

(2.4) $I(\mathcal{C})=I(\alpha,\beta,[\mathcal{C}])\in\mathbb{Z}.$

We let $\mathcal{M}^{J}(\alpha,\beta)$ denote the moduli space of
$J$-holomorphic currents with positive orbit set $\alpha$ and negative orbit
set $\beta$. Given $Z\in H_{2}(Y,\alpha,\beta)$, we define

$\mathcal{M}^{J}(\alpha,\beta,Z)=\left\\{\mathcal{C}\in\mathcal{M}^{J}(\alpha,\beta)\;\big{|}\;[\mathcal{C}]=Z\right\\}.$

The starting point for the foundations of ECH is the following “index
inequality”, originally from [29], see also [33, §3.4]:

######  Proposition 2.5.

If $C\in\mathcal{M}^{J}(\alpha,\beta)$ is somewhere injective, then

(2.5) $\operatorname{ind}(C)\leq I(C),$

with equality only if $C$ is embedded in $\mathbb{R}\times Y$.

Here $\operatorname{ind}(C)$ denotes the Fredholm index of $C$, which equals
the dimension of the moduli space of $J$-holomorphic curves near $C$ when $J$
is generic; see e.g. [33, §3.2] for details.

Using the index inequality, one can prove the following key result:

######  Proposition 2.6.

[33, Prop. 3.7] Suppose $J$ is a generic $\lambda$-compatible almost complex
structure. Let $\alpha$ and $\beta$ be orbit sets and let
$\mathcal{C}\in\mathcal{M}^{J}(\alpha,\beta)$. Then:

  * (0)

$I(\mathcal{C})\geq 0$, with equality if and only if $\mathcal{C}$ is a sum of
trivial cylinders with multiplicities.

  * (1)

If $I(\mathcal{C})=1$, then $\mathcal{C}=\mathcal{C}_{0}+C_{1}$, where
$I(\mathcal{C}_{0})=0$, and $C_{1}$ is embedded and disjoint from
$\mathcal{C}_{0}$ and satisfies $\operatorname{ind}(C_{1})=I(C_{1})=1$.

  * (2)

If $I(\mathcal{C})=2$, and if $\alpha$ and $\beta$ are ECH generators, then
$\mathcal{C}=\mathcal{C}_{0}+C_{1}$, where $I(\mathcal{C}_{0})=0$, and $C_{1}$
is embedded and disjoint from $\mathcal{C}_{0}$ and satisfies
$\operatorname{ind}(C_{1})=I(C_{1})=2$.

###  2.3. The definition of ECH

Let $\Gamma\in H_{1}(Y)$, and let $\lambda$ be a nondegenerate contact form on
$Y$. The embedded contact homology $ECH(Y,\lambda,\Gamma)$ is a vector space
over $\mathbb{Z}/2$ defined as follows. It is also possible to define ECH with
integer coefficients, as explained in [35, §9], but we will not need this.

######  Definition 2.7.

We define the chain complex $ECC(Y,\lambda,\Gamma)$ to be the $\mathbb{Z}/2$
vector space generated by ECH generators $\alpha$ with $[\alpha]=\Gamma$.
Next, choose a generic $\lambda$-compatible almost complex structure $J$. We
define a differential

$\partial_{J}:ECC(Y,\lambda,\Gamma)\longrightarrow ECC(Y,\lambda,\Gamma)$

as follows. If $\alpha$ is an ECH generator with $[\alpha]=\Gamma$, then

$\partial_{J}\alpha=\sum_{\beta}\langle\partial_{J}\alpha,\beta\rangle\beta$

where

$\langle\partial_{J}\alpha,\beta\rangle=\\#\\{\mathcal{C}\in\mathcal{M}^{J}(\alpha,\beta)/\mathbb{R}\mid
I(\mathcal{C})=1\\}.$

Here the sum is over ECH generators $\beta$ with $[\beta]=\Gamma$, and $\\#$
denotes the mod 2 count.

A compactness argument in [33, §5.3] shows that $\partial_{J}$ is well
defined. It is shown in [34, 35] that $\partial_{J}^{2}=0$. We then define
$ECH(Y,\lambda,\Gamma)$ to be the homology of the chain complex
$(ECC(Y,\lambda,\Gamma),\partial_{J})$.

It follows from the index ambiguity formula (2.3) that the ECH index defines a
relative $\mathbb{Z}/d$ grading on $ECH(Y,\lambda,\Gamma)$, where $d$ is the
divisibility of $c_{1}(\xi)+2\operatorname{PD}$ in $H^{2}(Y;\mathbb{Z})$ mod
torsion, which is always an even integer. Here the index difference between
two ECH generators $\alpha$ and $\beta$ with $[\alpha]=[\beta]=\Gamma$ is
defined to be $I(\alpha,\beta,Z)\mod d$ for any $Z\in H_{2}(Y,\alpha,\beta)$.
We say that a class in $ECH(Y,\lambda,\Gamma)$ is homogeneous if it can be
represented by a sum of ECH generators each of which have the same relative
$\mathbb{Z}/d$-grading. In the special case $\Gamma=0$, there is a canonical
$\mathbb{Z}/d$ grading, where the grading of an ECH generator $\alpha$ with
$[\alpha]=0$ is defined to be

(2.6) $I(\alpha)=I(\alpha,\emptyset,Z)$

for any $Z\in H_{2}(Y,\alpha,\emptyset)$. By [29, Prop. 1.6(c)], there is also
a canonical $\mathbb{Z}/2$ grading on $ECH(Y,\lambda,\Gamma)$ for all
$\Gamma$; an ECH generator $\alpha=\\{(\alpha_{i},m_{i})\\}$ has even grading
if and only if an even number of the Reeb orbits $\alpha_{i}$ are positive
hyperbolic.

Taubes [49] proved that if $Y$ is connected, then there is a canonical
isomorphism of relatively $\mathbb{Z}/d$-graded $\mathbb{Z}/2$-vector spaces

(2.7)
$ECH_{*}(Y,\lambda,\Gamma)\simeq\widehat{HM}^{-*}(Y,\mathfrak{s}_{\xi}+\operatorname{PD}(\Gamma);\mathbb{Z}/2).$

Here the right hand side is a version of Seiberg-Witten Floer cohomology, as
defined by Kronheimer-Mrowka [40], with $\mathbb{Z}/2$-coefficients. Also
$\mathfrak{s}_{\xi}$ denotes the spin-c structure on $Y$ determined by the
oriented $2$-plane field $\xi$; the spin-c structure
$\mathfrak{s}_{\xi}+\operatorname{PD}(\Gamma)$ has first Chern class
$c_{1}(\xi)+2\operatorname{PD}(\Gamma)$. In particular, it follows from the
canonical isomorphism (2.7) that $ECH(Y,\lambda,\Gamma)$ does not depend on
the choice of generic $\lambda$-compatible almost complex structure $J$, and
so we can denote it by $ECH(Y,\xi,\Gamma)$.

###  2.4. The $U$ map

Suppose now that the three-manifold $Y$ is connected. There is then a
“$U$-map”

(2.8) $U:ECH_{*}(Y,\xi,\Gamma)\longrightarrow ECH_{*-2}(Y,\xi,\Gamma)$

which will play a key role in our arguments.

######  Definition 2.8.

Let $y$ be a point on $Y$ which is not on the image of any Reeb orbit, and let
$J$ be a generic $\lambda$-compatible almost complex structure. We then define
a map

$U_{J,y}:ECC_{*}(Y,\lambda,\Gamma)\longrightarrow ECC_{*-2}(Y,\lambda,\Gamma)$

as follows. If $\alpha$ is an ECH generator with $[\alpha]=\Gamma$, then

$U_{J,y}\alpha=\sum_{\beta}\langle U_{J,y}\alpha,\beta\rangle\beta$

where

(2.9) $\langle
U_{J,y}\alpha,\beta\rangle=\\#\left\\{\mathcal{C}\in\mathcal{M}^{J}(\alpha,\beta)\mid
I(\mathcal{C})=2,\;(0,y)\in\mathcal{C}\right\\}.$

As explained in [36, §2.5], $U_{J,y}$ is a chain map:

$\partial_{J}\circ U_{J,y}=U_{J,y}\circ\partial_{J}.$

Consequently $U_{J,y}$ induces a map on $ECH(Y,\xi,\Gamma)$. As shown in [36,
§2.5], this map does not depend on the choice of point $y$ (here one uses the
assumption that $Y$ is connected), so we can denote it by $U_{J}$. Taubes [50]
further showed that under the isomorphism (2.7), the map $U_{J}$ agrees with a
“U-map” on Seiberg-Witten Floer cohomology, so in particular we have a well-
defined map as in (2.8).

We will need the following fact about the nontriviality of the $U$-map, which
is a consequence of [9, Lem. A.1(a)].

######  Definition 2.9.

A $U$-sequence in $ECH(Y,\xi,\Gamma)$ is a sequence of nonzero classes
$\\{\sigma_{k}\\}_{k\geq k_{0}}$ in $ECH(Y,\xi,\Gamma)$ such that
$U\sigma_{k+1}=\sigma_{k}$ for each $k\geq k_{0}$.

######  Proposition 2.10.

If $Y$ is connected and $c_{1}(\xi)+2\operatorname{PD}(\Gamma)\in
H^{2}(Y;\mathbb{Z})$ is torsion, then there exists a $U$-sequence
$\\{\sigma_{k}\\}_{k\geq k_{0}}$ in $ECH(Y,\xi,\Gamma)$ such that each class
$\sigma_{k}$ is homogeneous and has even grading.

###  2.5. Spectral invariants and the volume property

It follows from the definition of “$\lambda$-compatible” almost complex
structure that the differential $\partial_{J}$, and the U-map $U_{J,y}$,
decrease the symplectic action: If $\alpha$ and $\beta$ are ECH generators
with $\langle\partial_{J}\alpha,\beta\rangle$ or $\langle
U_{J,y}\alpha,\beta\rangle$ nonzero, then
$\mathcal{A}(\alpha)>\mathcal{A}(\beta)$. In particular, for each real number
$L$, we can define the filtered embedded contact homology, denoted by
$ECH^{L}(Y,\lambda,\Gamma)$, to be the homology of the subcomplex of
$(ECC(Y,\lambda,\Gamma),\partial_{J})$ generated by ECH generators with
symplectic action less than $L$. There is also an inclusion-induced map

(2.10) $ECH^{L}(Y,\lambda,\Gamma)\longrightarrow ECH(Y,\xi,\Gamma).$

It is shown in [37, Thm. 1.3] that the filtered ECH and the inclusion-induced
map (2.10) do not depend on the choice of generic $\lambda$-compatible almost
complex structure $J$. However the filtered ECH, unlike the regular ECH, does
depend on the contact form $\lambda$ and not just on the contact structure
$\xi$, as detected for example by the following spectral invariants.

######  Definition 2.11.

Given a nonzero class $\sigma\in ECH(Y,\xi,\Gamma)$, we define the spectral
invariant

$c_{\sigma}(Y,\lambda)\in\mathbb{R}$

to be the infimum over $L$ such that $\sigma$ is in the image of the
inclusion-induced map (2.10).

Equivalently, if one makes a choice of generic $\lambda$-compatible $J$, then
$c_{\sigma}(Y,\lambda)$ is the minimum $L$ such that the class $\sigma$ can be
represented by a cycle in the chain complex
$(ECC(Y,\lambda,\Gamma),\partial_{J})$ which is a sum of ECH generators each
having symplectic action less than or equal to $L$. As shown in [32], each
spectral invariant $c_{\sigma}$ is a $C^{0}$-continuous function of the
contact form $\lambda$, and as such extends by continuity to degenerate
contact forms.

The following “Weyl law” for the ECH spectrum will play a key role in the
proof of the main theorem. This is a special case of [10, Thm. 1.3].

######  Proposition 2.12.

Suppose $Y$ is connected and $c_{1}(\xi)+2\operatorname{PD}(\Gamma)\in
H^{2}(Y;\mathbb{Z})$ is torsion. Let $\\{\sigma_{k}\\}_{k\geq k_{0}}$ be a
$U$-sequence in $ECH(Y,\xi,\Gamma)$. Then

$\lim_{k\to\infty}\frac{c_{\sigma_{k}}(Y,\lambda)^{2}}{k}=2\int_{Y}\lambda\wedge
d\lambda.$

###  2.6. Partition conditions

Let $\alpha=\\{(\alpha_{i},m_{i})\\}$ and $\beta=\\{(\beta_{j},n_{j})\\}$ be
orbit sets, and suppose that $C\in\mathcal{M}^{J}(\alpha,\beta)$ is a
somewhere injective curve. Then $C$ has positive ends at covers of
$\alpha_{i}$ with total multiplicity $m_{i}$; the multiplicities of these
covers determine a partition of $m_{i}$, which we denote by
$p_{\alpha_{i}}^{+}(C)$. Likewise, the multiplicities of the covers of
$\beta_{j}$ at which $C$ has negative ends determine a partition of $n_{j}$,
which we denote by $p_{\beta_{j}}^{-}(C)$. It turns out that if equality holds
in the index inequality (2.5), then these partitions are uniquely determined
as follows.

######  Definition 2.13.

If $\theta$ is a real number and $m$ is a positive integer, we define two
partitions of $m$, the positive partition $p_{\theta}^{+}(m)$ and the negative
partition $p_{\theta}^{-}(m)$, as follows. We define the positive partition
$p^{+}_{\theta}(m)$ to be the set of horizontal displacements of the maximal
polygonal path that starts at $(0,0)$, ends at $(m,\lfloor m\theta\rfloor)$,
is the graph of a piecewise linear concave function with vertices at integer
lattice points, and does not go above the line $y=m\theta$. Here “horizontal
displacement” means the horizontal displacement between two consecutive
integer lattice points on the path. Similarly, we define the negative
partition $p^{-}_{\theta}(m)$ to be the set of horizontal displacements of the
edges of the minimal polygonal path that starts at $(0,0)$, ends at $(m,\lceil
m\theta\rceil)$, is the graph of a piecewise linear convex function with
vertices at integer lattice points, and does not go below the line
$y=m\theta$.

Note that $p_{\theta}^{-}(m)=p_{-\theta}^{+}(m)$. Also note that for a given
$m$, the partitions $p_{\theta}^{+}(m)$ and $p_{\theta}^{-}(m)$ depend only on
the equivalence class of $\theta$ in $\mathbb{R}/\mathbb{Z}$. The following is
a basic example:

######  Example 2.14.

If $0\leq\theta<1/m$, then $p_{\theta}^{+}(m)=(1,\ldots,1)$. This is because
the maximal polygonal path described above is the horizontal line from $(0,0)$
to $(m,0)$.

We will also need the following generalization:

######  Example 2.15.

If $\theta$ is a rational number $a/b$ in lowest terms, and if $m$ is a
multiple of $b$, then for $\epsilon\geq 0$ sufficiently small with respect to
$m$ and $\theta$, we have $p_{a/b+\varepsilon}^{+}(m)=(m/b,\ldots,m/b)$. This
is because the maximal polygonal path described above is the line from $(0,0)$
to $(m,ma/b)$.

If $\gamma$ is a simple Reeb orbit, then the linearized Reeb flow has a well-
defined rotation number $\theta\in\mathbb{R}/\mathbb{Z}$ independent of a
choice of trivialization. Given a positive integer $m$, we define

$p_{\gamma}^{+}(m)=p_{\theta}^{+}(m),\quad\quad\quad
p_{\gamma}^{-}(m)=p_{\theta}^{-}(m).$

Now let $J$ be a generic $\lambda$-compatible almost complex structure,
suppose that $\mathcal{C}\in\mathcal{M}^{J}(\alpha,\beta)$ has ECH index $1$
or $2$, and in the latter case suppose that $\alpha$ and $\beta$ are ECH
generators. Write $\mathcal{C}=\mathcal{C}_{0}+C_{1}$ as in Proposition 2.6.
For each $i$, let $m_{i}^{\prime}$ denote the total covering multiplicity of
the covers of $\alpha_{i}$ at which $C_{1}$ has positive ends. Thus
$m_{i}^{\prime}\leq m_{i}$, and $m_{i}-m_{i}^{\prime}$ is the multiplicity of
the trivial cylinder $\mathbb{R}\times\alpha_{i}$ in $\mathcal{C}_{0}$.
Likewise, let $n_{j}^{\prime}$ denote the total covering multiplicity of the
covers of $\beta_{j}$ at which $C_{1}$ has negative ends.

######  Proposition 2.16.

Let $J$ be generic, let $\mathcal{C}\in\mathcal{M}^{J}(\alpha,\beta)$ be a
$J$-holomorphic current with ECH index $1$ or $2$, assume that $\alpha$ and
$\beta$ are ECH generators if $I(\mathcal{C})=2$, and write
$\mathcal{C}=\mathcal{C}_{0}+C_{1}$ as in Proposition 2.6. Then:

  * •

For each $i$ with $m_{i}^{\prime}>0$, and for each $n$ with $0\leq n\leq
m_{i}-m_{i}^{\prime}$, we have

(2.11) $p_{\alpha_{i}}^{+}(m_{i}^{\prime}+n)=p_{\alpha_{i}}^{+}(C_{1})\cup
p_{\alpha_{i}}^{+}(n).$

  * •

Similarly, for each $j$ with $n_{j}^{\prime}>0$, and for each $n$ with $0\leq
n\leq n_{j}-n_{j}^{\prime}$, we have

(2.12) $p_{\beta_{j}}^{-}(n_{j}^{\prime}+n)=p_{\beta_{j}}^{-}(C_{1})\cup
p_{\beta_{j}}^{-}(n).$

Here $\cup$ denotes the union of multisets, and we interpret
$p_{\gamma}^{\pm}(0)=\emptyset$.

In particular, (2.11) implies that if $C_{1}$ has positive ends at covers of
$\alpha_{i}$, then the multiplicities of these covers are determined by
$p_{\alpha_{i}}^{+}(C_{1})=p_{\alpha_{i}}^{+}(m_{i}^{\prime})$.

######  Remark 2.17.

In the case when $\mathcal{C}_{0}$ includes the trivial cylinder
$\mathbb{R}\times\alpha_{i}$ and $C_{1}$ has a positive end asymptotic to a
cover of $\alpha_{i}$, so that $m_{i}>m_{i}^{\prime}>0$, it follows from
(2.11) with $n=m_{i}-m_{i}^{\prime}$ that if $1\in
p_{\alpha_{i}}^{+}(m_{i}-m_{i}^{\prime})$, then $1\in
p_{\alpha_{i}}^{+}(m_{i})$. The converse is also true; this follows from [34,
Lem. 7.28]. Likewise, if $n_{j}>n_{j}^{\prime}>0$, then $1\in
p_{\beta_{j}}^{-}(n_{j}-n_{j}^{\prime})$ if and only if $1\in
p_{\beta_{j}}^{-}(n_{j})$.

Here are some additional facts about the partitions that we will need.

######  Lemma 2.18.

Let $\theta\in\mathbb{R}$ be irrational, and let $m$ be a positive integer.
Then:

  * (a)

$1\in p_{\theta}^{\pm}(m)\Longleftrightarrow 1\notin p_{\theta}^{\mp}(m)$.

  * (b)

If $m>1$, then the partitions $p_{\theta}^{+}(m)$ and $p_{\theta}^{-}(m)$ are
disjoint.

  * (c)

If $|p_{\theta}^{+}(m)|+|p_{\theta}^{-}(m)|\leq 3$, then $m\\{\theta\\}<2$ or
$m(1-\\{\theta\\})<2$, where $\\{\theta\\}\in[0,1)$ denotes the fractional
part of $\theta$.

###### Proof.

Statements (a) and (b) are proved in [29, Rem. 4.4]; see also [33, Ex.
3.13(c)] for a stronger version of (b). Statement (c) is shown in the proof of
[9, Lem. 4.9]. ∎

###  2.7. The $J_{0}$ index and topological complexity bounds

Recall from the index inequality (2.5) that the ECH index bounds the Fredholm
index of a somewhere injective curve. There is a variant of the ECH index,
called the $J_{0}$-index, which instead bounds the topological complexity of
the curve.

######  Definition 2.19.

Let $\alpha=\\{(\alpha_{i},m_{i})\\}$ and $\beta=\\{(\beta_{j},n_{j})\\}$ be
orbit sets with $[\alpha]=[\beta]$, and let $Z\in H_{2}(Y,\alpha,\beta)$. We
define

(2.13) $J_{0}(\alpha,\beta,Z)=I(\alpha,\beta,Z)-\left(2c_{\tau}(Z)+{\rm
CZ}_{\tau}^{\operatorname{top}}(\alpha)-{\rm
CZ}_{\tau}^{\operatorname{top}}(\beta)\right).$

Here ${\rm CZ}_{\tau}^{\operatorname{top}}(\alpha)$ adds up the Conley-Zehnder
indices of the “top” iterates of the orbits $\alpha_{i}$, namely

(2.14) ${\rm CZ}_{\tau}^{\operatorname{top}}(\alpha)=\sum_{i}{\rm
CZ}_{\tau}(\alpha_{i}^{m_{i}}).$

If $\mathcal{C}\in\mathcal{M}^{J}(\alpha,\beta)$, then similarly to (2.4), we
write $J_{0}(\mathcal{C})=J_{0}(\alpha,\beta,[\mathcal{C}])$.

In the situation relevant to us, $J_{0}$ bounds topological complexity as
follows. Suppose that the almost complex structure $J$ is generic, let
$\alpha$ and $\beta$ be ECH generators, suppose
$\mathcal{C}\in\mathcal{M}^{J}(\alpha,\beta)$ has ECH index $2$, and write
$\mathcal{C}=\mathcal{C}_{0}+C_{1}$ as in Proposition 2.6. For each simple
Reeb orbit $\gamma$, define $e_{\gamma}^{+}(\mathcal{C})$ to be twice the
number of positive ends of $C_{1}$ at covers of $\gamma$, minus $1$ if $C_{1}$
has positive ends at covers of $\gamma$ but $\mathcal{C}_{0}$ does not include
the trivial cylinder $\mathbb{R}\times\gamma$. Likewise, define
$e_{\gamma}^{-}(\mathcal{C})$ to be the twice the number of negative ends of
$C_{1}$ at covers of $\gamma$, minus $1$ if $C_{1}$ has negative ends at
covers of $\gamma$ but $\mathcal{C}_{0}$ does not include the trivial cylinder
$\mathbb{R}\times\gamma$. Write
$e_{\gamma}(\mathcal{C})=e_{\gamma}^{+}(\mathcal{C})+e_{\gamma}^{-}(\mathcal{C})$.
We then have:

######  Proposition 2.20.

In the above situation we have

(2.15) $J_{0}(\mathcal{C})=2g(C_{1})-2+\sum_{\gamma}e_{\gamma}(\mathcal{C}).$

Here $g(C_{1})$ denotes the genus of $C_{1}$, and the sum is over simple Reeb
orbits $\gamma$.

###### Proof.

The fact that the left hand side is greater than or equal to the right hand
side is a restatement of [36, Lem. 3.5]. Although this inequality is all that
we actually need, one can further obtain equality as explained in the footnote
on [36, page 919]. ∎

###  2.8. Criterion for a Birkhoff section

We now recall a criterion for when the nontrivial component of a holomorphic
curve counted by the $U$ map gives rise to a Birkhoff section for the Reeb
flow.

######  Definition 2.21.

Let $(Y,\lambda)$ be a contact three-manifold. A Birkhoff section for the Reeb
flow is an embedded open surface $\Sigma\subset Y$ such that:

  * •

The Reeb vector field $R$ is transverse to $\Sigma$.

  * •

There is a compact surface with boundary $\overline{\Sigma}$, such that
$\operatorname{int}(\overline{\Sigma})=\Sigma$, and the inclusion $\Sigma\to
Y$ extends to an immersion $g:\overline{\Sigma}\to Y$ such that each boundary
circle of $\overline{\Sigma}$ is mapped to the image of a Reeb orbit, and
$\Sigma\cap g\left(\partial\overline{\Sigma}\right)=\emptyset$.

  * •

For every $y\in Y\setminus g\left(\partial\overline{\Sigma}\right)$, the Reeb
trajectory starting at $y$ intersects $\Sigma$ in both forward and backward
time.

######  Proposition 2.22.

Let $J$ be a generic $\lambda$-compatible almost complex structure, let
$\alpha$ and $\beta$ be ECH generators, let
$\mathcal{C}\in\mathcal{M}^{J}(\alpha,\beta)$, and suppose that
$I(\mathcal{C})=2$. Write $\mathcal{C}=\mathcal{C}_{0}+C_{1}$ as in
Proposition 2.6. Suppose that:

  * (a)

$C_{1}$ has at least two ends.

  * (b)

$C_{1}$ has genus zero.

  * (c)

$C_{1}$ does not have two positive ends, or two negative ends, at covers of
the same simple Reeb orbit.

  * (d)

$C_{1}$ has no ends at positive hyperbolic orbits.

  * (e)

The component of the moduli space $\mathcal{M}^{J}(\alpha,\beta)/\mathbb{R}$
containing $C_{1}$ is compact.

Then the restriction of the projection $\mathbb{R}\times Y\to Y$ to $C_{1}$ is
an embedding, and its image is a Birkhoff section for the Reeb flow.

###### Proof.

This follows from [9, Prop. 3.2], as explained in [9, §5.2]. ∎

##  3\. Birkhoff sections for nondegenerate perturbations

###  3.1. Introduction

The aim of this section is to prove the following proposition.

######  Proposition 3.1.

Let $Y$ be a closed connected three-manifold, let $\lambda$ be a contact form
on $Y$ with finitely many simple Reeb orbits, and assume that $c_{1}(\xi)\in
H^{2}(Y;\mathbb{Z})$ is torsion, where $\xi=\ker\lambda$. Let $\eta>0$ be
given. Then there exists a sequence
$(\lambda_{k},J_{k},\alpha_{k},\beta_{k},C_{k})_{k=1,2,\dots}$ where:

  * •

$\lambda_{k}$ is a nondegenerate contact form, and $\lambda_{k}\to\lambda$ in
$C^{\infty}$ as $k\to\infty$,

  * •

$J_{k}$ is a $\lambda_{k}$-compatible almost complex structure, and $J_{k}\to
J$ in $C^{\infty}$ as $k\to\infty$, where $J$ is a $\lambda$-compatible almost
complex structure,

  * •

$\alpha_{k}$ and $\beta_{k}$ are (not necessarily simple) Reeb orbits for
$\lambda_{k}$ such that $\alpha_{k}\to\alpha_{\infty}$ and
$\beta_{k}\to\beta_{\infty}$ in $C^{\infty}$, where $\alpha_{\infty}$ and
$\beta_{\infty}$ are Reeb orbits for $\lambda$,

  * •

$C_{k}$ is a $J_{k}$-holomorphic cylinder in $\mathbb{R}\times Y$ with
positive end asymptotic to $\alpha_{k}$ and negative end asymptotic to
$\beta_{k}$ for each $k$,

such that for each $k$:

  1. (a)

$\operatorname{ind}(C_{k})=I(C_{k})=2$, see §2.2.

  2. (b)

$\mathcal{A}(\alpha_{k})-\mathcal{A}(\beta_{k})<\eta$.

  3. (c)

If $\alpha_{\infty}$ is degenerate, then the rotation number of $\alpha_{k}$
is in the interval $(-1/2,0)$. In particular this rotation number converges to
$0$ from below as $k\to\infty$. Similarly, if $\beta_{\infty}$ is degenerate,
then the rotation number of $\beta_{k}$ is in the interval $(0,1/2)$.

  4. (d)

$C_{k}$ projects to a Birkhoff section for the Reeb flow for $\lambda_{k}$,
see §2.8.

######  Remark 3.2.

This is a variant of [9, Prop. 4.2], which for a single nondegenerate contact
form with only finitely many simple Reeb orbits found a genus zero Birkhoff
section with at most three punctures. For our purposes it is important to
obtain a sequence of cylinders with uniform bounds on the action as
$k\to\infty$.

######  Remark 3.3.

As explained in [36, §4.6], the existence of a Birkhoff section given by
holomorphic cylinders implies that the three-manifold $Y$ is a lens space or
$S^{3}$. Hence Proposition 3.1 already implies Theorem 1.1 when $Y$ is not
$S^{3}$ or a lens space; in this case $\lambda$ must have infinitely many
simple Reeb orbits.

#### Outline of the proof of Proposition 3.1

For any given $\lambda_{k}$, one can use the non-triviality of the $U$-map in
ECH to produce a “chain” of a large number of $J_{k}$-holomorphic curves with
ECH index $2$. The first challenge is to show that some of these are
cylinders. Building on ideas from [36, 9], we accomplish this by a kind of
probabilistic argument, showing that in fact “most” of the curves must be
cylinders. The starting point behind is to show that the curves on average
have $J_{0}$ approximately $2$; this is the one place where we use the
assumption that $c_{1}(\xi)$ is torsion. A $J_{0}=2$ curve need not be a
cylinder, however (see §2.7), and so a deeper analysis is needed. The main new
ingredient for this is to define a “score” for each curve; roughly speaking,
the idea is that cylinders should be characterized by having score $0$, and on
the other hand the score is nonnegative, so, one can deduce the desired
existence result by arguing that the total score of the chain of curves is not
too large.

Since we must consider a sequence of contact forms $\lambda_{k}$, we must also
keep careful track of the various parameters in these probabilistic arguments.
In particular, we will define a $k$-independent constant $T_{q}$ such that all
holomorphic curves under consideration go between orbit sets for $\lambda_{k}$
with action $\leq T_{q}$.

As a result, the proof of Proposition 3.1 shows the following more general
statement, which is also applicable to contact forms with infinitely many
simple Reeb orbits; as mentioned previously, it gives, for example,
information about the growth rate of the orbits when the manifold is not a
lens space and the contact structure is torsion. (For simplicity we will not
write a separate proof of this as it is not needed for the main theorem.)

######  Proposition 3.4.

Let $Y$ be a closed connected three-manifold, let $\lambda$ be a contact form
on $Y$, and assume that $c_{1}(\xi)\in H^{2}(Y;\mathbb{Z})$ is torsion, where
$\xi=\operatorname{Ker}(\lambda)$. Suppose that $\lambda$ has at least $n$
simple Reeb orbits. Let $\gamma_{1},\ldots,\gamma_{n}$ denote the first $n$
simple Reeb orbits in terms of action222That is,
$\mathcal{A}(\gamma_{1})\leq\cdots\leq\mathcal{A}(\gamma_{n})$, and any other
simple Reeb orbit has action at least that of $\gamma_{n}$.. Let $\eta>0$ be
given. Then there is a constant $T$ computable in terms of $\eta$, the ECH
spectral invariants of $\lambda$, and the actions, rotation numbers, and
homology clases of $\gamma_{1},\ldots,\gamma_{n}$, such that the following
holds: If $\lambda$ does not have any simple Reeb orbit of action $\leq T$
other than $\gamma_{1},\ldots,\gamma_{n}$, then there exists a sequence
$(\lambda_{k},J_{k},\alpha_{k},\beta_{k},C_{k})_{k=1,2,\dots}$ as in
Proposition 3.1 satisfying all of the conclusions of Proposition 3.1.

###  3.2. Setting the stage

We first collect various preliminaries results, estimates, and constants that
will be used in our arguments. A main theme is to extract constants that are
independent of sufficiently large $k$.

####  3.2.1. The perturbation lemma

The sequence of contact forms in Proposition 3.1 will come from the following
lemma, the idea of which is due to Bangert [2]:

######  Lemma 3.5.

Let $(Y,\lambda)$ be a closed contact three-manifold with finitely many simple
Reeb orbits. Let $N$ be the union of pairwise disjoint open tubular
neighborhoods of these orbits. Then there is a sequence of nondegenerate
contact forms $(\lambda_{k}=f_{k}\lambda)_{k=1,2,\ldots}$ converging in
$C^{\infty}$ to $\lambda$, such that:

  1. (a)

For every $k$ and every simple Reeb orbit $\gamma$ of $\lambda$, we have
$f_{k}|_{\gamma}=1$ and $df_{k}|_{\gamma}=0$. In particular, $\gamma$ is also
a simple Reeb orbit of $\lambda_{k}$.

  2. (b)

There exists a compact neighborhood $K\subset N$ of the Reeb orbits of
$\lambda$ such that $f_{k}|_{Y\setminus K}=1$ for every $k$.

  3. (c)

For every $T>0$ there exists $k_{T}$ such that if $k\geq k_{T}$ then each
simple Reeb orbit $\gamma$ for $\lambda_{k}$ with action $\leq T$ is contained
in $N$, and for each such $\gamma$, there is a simple Reeb orbit
$\gamma^{\prime}$ for $\lambda$ in the same component of $N$ such that:

    1. (1)

If $\gamma^{\prime}$ and all its iterates are nondegenerate, then
$\gamma=\gamma^{\prime}$.

    2. (2)

If $\gamma^{\prime}$ has rational rotation number, written in lowest terms
with respect to some trivialization as $a/b$ with $b>0$, then either
$\gamma=\gamma^{\prime}$, or $\gamma$ is $C^{\infty}$-close to the $b$-fold
cover of $\gamma^{\prime}$.

In particular, for any $\varepsilon>0$, if $k$ is sufficiently large, then the
action and the rotation number of $\gamma$ are within $\varepsilon$ of those
for $\gamma^{\prime}$ in case (1), and $(\gamma^{\prime})^{b}$ in case (2).

###### Proof.

Consider a sequence of nondegenerate contact forms $\lambda_{k}=f_{k}\lambda$
such that $f_{k}\to 1$ in $C^{\infty}$, and such that on each of the finitely
many simple Reeb orbits $\gamma$ of $\lambda$ we have $f_{k}|_{\gamma}=1$ and
$df_{k}|_{\gamma}=0$. In particular, each $\gamma$ is a simple Reeb orbit for
$\lambda_{k}$ with the same period. We can further assume that $f_{k}=1$
outside of a small compact neighborhood $K\subset N$ of the Reeb orbits of
$\lambda$. All the desired conclusions now follow from Bangert’s lemma as
stated in [7, Lemma 3.8] or in [2, Proposition 1]. ∎

######  Remark 3.6.

In part (c), when $\gamma^{\prime}$ is degenerate, there may be many simple
Reeb orbits $\gamma$ of $\lambda_{k}$ associated to it.

####  3.2.2. Setting up the proof of Proposition 3.1

Let $(Y,\lambda)$ and $\eta>0$ as in Proposition 3.1 be given. Throughout the
rest of this section, fix a sequence of contact forms $\lambda_{k}$ as
provided by Lemma 3.5. Let $J$ be any $\lambda$-compatible almost complex
structure on $\mathbb{R}\times Y$. We can find a sequence $J_{k}$ of
$\lambda_{k}$-compatible almost complex structures on $\mathbb{R}\times Y$
that converges in $C^{\infty}$ to $J$, and we can arrange that $J_{k}$ is
generic as needed to define the ECH of $\lambda_{k}$.

Let $\mathcal{P}_{k}$ denote the set of simple Reeb orbits of $\lambda_{k}$.
An orbit set $\alpha$ for $\lambda_{k}$ is then equivalent to a finitely
supported “multiplicity” function $\mathcal{P}_{k}\to\mathbb{Z}_{\geq 0}$. The
orbit set $\alpha=\\{(\alpha_{i},m_{i})\\}$ corresponds to the function
sending $\alpha_{i}\mapsto m_{i}$ and sending all other simple Reeb orbits of
$\lambda_{k}$ to zero. We denote the multiplicity function evaluated on a
simple Reeb orbit $\gamma\in\mathcal{P}_{k}$ by $m(\gamma,\alpha)$. We refer
to the numbers $m(\gamma,\alpha)$ as “components” of $\alpha$. Define the
complexity of $\alpha$ to be the maximum of the components $m(\gamma,\alpha)$,
and denote the number of simple Reeb orbits that appear in $\alpha$ by
$|\alpha|=\left|\\{\gamma\in\mathcal{P}_{k}\mid m(\gamma,\alpha)>0\\}\right|$.
Likewise let $\mathcal{P}$ denote the set of simple Reeb orbits of $\lambda$;
then orbit sets for $\lambda$ correspond to finitely supported functions
$\mathcal{P}\to\mathbb{Z}_{\geq 0}$, and we define the complexity analogously.

Fix $N$ as in Lemma 3.5, and fix a symplectic trivialization $\tau$ of $\xi$
over all simple Reeb orbits of $\lambda$. This induces trivializations over
all (not necessarily simple) Reeb orbits of $\lambda$. Moreover, once an
action bound $T>0$ is fixed, if we asssume that $k\geq k_{T}$, where $k_{T}$
is the constant given by Lemma 3.5, then $\tau$ induces trivializations, again
denoted by $\tau$, of all Reeb orbits of $\lambda_{k}$ of action $\leq T$.
This is because the trivialization $\tau$ induces a symplectic trivialization,
uniquely up to homotopy, of the contact structure $\xi$ over $N$. In addition,
in this situation, if $\alpha$ is an orbit set for $\lambda_{k}$, then there
is a unique orbit set $\underline{\alpha}$ for which is homotopic to $\alpha$
in $N$.

Fix a point $y\in Y\setminus N$ to define the chain map $U_{J_{k},y}$, see
§2.4. We define a $U$-curve to be a current counted by this chain map, namely
a current $\mathcal{C}\in\mathcal{M}^{J_{k}}(\alpha,\beta)$ with
$(0,y)\in\mathcal{C}$, where $\alpha$ and $\beta$ are ECH generators for
$\lambda_{k}$, and the ECH index $I(\mathcal{C})=2$.

####  3.2.3. Basic estimates

The following estimate gives a useful lower bound on the difference of actions
between certain orbit sets.

######  Lemma 3.7.

Let $M>1$. Then there is a constant $\epsilon_{M}>0$ such that for any two
orbit sets $\alpha$, $\beta$ for $\lambda$ with distinct actions, if either
orbit set has complexity $\leq M$, or if both orbit sets have exactly one
simple Reeb orbit with multiplicity $>M$ and these simple orbits are the same,
then

(3.1) $\left|\mathcal{A}(\alpha)-\mathcal{A}(\beta)\right|\geq\epsilon_{M}\,.$

###### Proof.

Let $S_{M}$ denote the set of actions of orbit sets of complexity $\leq M$.
Since the set $\mathcal{P}$ of simple Reeb orbits of $\lambda$ is finite, it
follows that the set $S_{M}$ is finite, and the set $S_{\infty}$ of all
actions of orbit sets is discrete. Thus there is a positive lower bound
$\epsilon^{\prime}_{M}$ on the set of distances between two distinct actions
of orbit sets with at least one in $S_{M}$.

Let $\alpha$ and $\beta$ be as in the statement of the lemma. If either orbit
set has complexity $\leq M$, then their actions will differ by at least
$\epsilon^{\prime}_{M}>0$.

If on the other hand both $\alpha$ and $\beta$ have exactly one simple Reeb
orbit with multiplicity $>M$, which is the same for both orbit sets, then
denote this simple Reeb orbit by $\gamma$. Consider the orbit sets
$\alpha_{0}$ and $\beta_{0}$ obtained from $\alpha$ and $\beta$ by changing
the multiplicity of $\gamma$ to zero and keeping the multiplicities of all
other simple Reeb orbits the same. Then $\alpha_{0}$ and $\beta_{0}$ are orbit
sets with complexity $\leq M$. Since the set $S_{M}$ is finite, it follows
that the set $S_{M}^{\prime}$ of differences between elements of $S_{M}$ is
also finite. Now we have

$\left|\mathcal{A}(\alpha)-\mathcal{A}(\beta)\right|=\left|\mathcal{A}(\alpha_{0})-\mathcal{A}(\beta_{0})+z\mathcal{A}(\gamma)\right|$

for some integer $z$. By finiteness of the set $S_{M}^{\prime}$, the right
hand side of the above equality, when nonzero, is bounded from below by some
$\epsilon_{M}^{\prime\prime}>0$. We then take
$\epsilon_{M}=\min(\epsilon_{M}^{\prime},\epsilon_{M}^{\prime\prime})$. ∎

There is also the following useful bound on the relative Chern class. If
$\alpha$ is a null-homologous orbit set (for $\lambda$ or $\lambda_{k}$),
define

$c_{\tau}(\alpha)=c_{\tau}(\alpha,\emptyset,Z)\in\mathbb{Z},$

where $c_{\tau}$ denotes the relative first Chern class defined in §2.1,
$\tau$ is the trivialization we fixed in §3.2.2, and $Z\in
H_{2}(Y,\alpha,\emptyset)$ is arbitrary. It follows from equation (2.2) and
the assumption that $c_{1}(\xi)$ is torsion that $c_{\tau}(\alpha)$ does not
depend on the choice of $Z$.

######  Lemma 3.8.

For each $\ell>0$ there exists $\delta_{1}>0$ (depending only on $\lambda$ and
$\ell$) such that if $\alpha$ is a nullhomologous orbit set for $\lambda$,
then

(3.2) $\left|2c_{\tau}(\alpha)+{\rm CZ}^{\rm
top}_{\tau}(\alpha))\right|+2\frac{\mathcal{A}({\alpha})}{\ell}\leq\delta_{1}\mathcal{A}(\alpha).$

Here ${\rm CZ}^{\operatorname{top}}$ is defined as in equation (2.14). In that
equation, we use the definition of the Conley-Zehnder index in equation (2.1)
also for degenerate Reeb orbits, as in [7].

###### Proof of Lemma 3.8..

Without the $2\mathcal{A}({\alpha})/\ell$ term, this follows from the same
argument as in [9, Lemma 4.3(b)]. The $2\mathcal{A}({\alpha})/\ell$ term can
be accommodated by increasing $\delta_{1}$. ∎

####  3.2.4. Exceptional pairs

Let $\gamma$ be a simple Reeb orbit for $\lambda_{k}$ and let $m$ be a
positive integer. As in [9], we call the pair $(\gamma,m)$ exceptional if
$|p^{+}_{\gamma}(m)|+|p^{-}_{\gamma}(m)|\leq 3$, where the partitions
$p_{\gamma}^{\pm}(m)$ are defined in §2.6. We will need the following variant
of [9, Lem. 4.9], restricting the exceptional pairs in a $k$-independent
manner:

######  Lemma 3.9.

There exists a constant $M\geq 3$, depending only on $\lambda$, such that the
following holds. For every $T>0$ there is a constant $k_{T}^{\prime}$ such
that if $k\geq k_{T}^{\prime}$, if $\gamma$ is a simple Reeb orbit for
$\lambda_{k}$, if $m\geq M$, and if $m\mathcal{A}(\gamma)\leq T$, then the
pair $(\gamma,m)$ is not exceptional.

###### Proof.

Given a number $\theta\in\mathbb{R}/\mathbb{Z}$, write

$\overline{\theta}=\min\\{\\{\theta\\},1-\\{\theta\\}\\}\in[0,1/2].$

By Lemma 2.18(c), if $\gamma\in\mathcal{P}_{k}$ has rotation number $\theta$,
and if $m\overline{\theta}>2$, then the pair $(\gamma,m)$ is not exceptional.
Also, by Example 2.14, if $m\overline{\theta}<1$ and $m\geq 3$ then the pair
$(\gamma,m)$ is not exceptional. So to prove the lemma, it is enough to show
that there exists a constant $M\geq 3$, depending only on $\lambda$, such
that:

  * (*)

For every $T>0$, there exists a constant $k_{T}^{\prime}$ such that if $k\geq
k_{T}^{\prime}$, if $\gamma\in\mathcal{P}_{k}$ has rotation number $\theta$,
if $m\geq M$, and if $m\mathcal{A}(\gamma)\leq T$, then
$m\overline{\theta}\notin(1,2)$.

Let $\Theta\subset\mathbb{R}/\mathbb{Z}$ denote the finite set consisting of
the rotation numbers of the simple Reeb orbits of $\gamma$. We claim that the
requirement (*) will be fulfilled as long as $M\geq 3$ and

(3.3)
$M>\max\left\\{2/\overline{\theta}\;\big{|}\;\theta\in\Theta\setminus\\{0\\}\right\\}.$

To see this, let $T>0$ be given. By Lemma 3.5, if $k$ is sufficiently large,
then the rotation number $\theta$ of a simple Reeb orbit $\gamma$ of
$\lambda_{k}$ with action less than $T$ will be close to $\Theta\cup\\{0\\}$.
If $\theta$ is close to $\Theta\setminus\\{0\\}$ and $m\geq M$, then we will
have $m\overline{\theta}>2$ by (3.3). If $\theta$ is close to $0$ and $m\geq
M$ and $m\mathcal{A}(\gamma)\leq T$, then we will have $m\overline{\theta}<1$,
because by Lemma 3.5, for $k$ sufficiently large we have $m\leq T/\ell+1$
where $\ell$ is the minimum action of a Reeb orbit of $\lambda$. ∎

####  3.2.5. Some constants

We now define some constants associated to $\lambda$ that will be used below.
We emphasize that these constants only depend on $\lambda$ and are independent
of $k$.

  * •

Let $B_{0}$ be the maximum of the denominators $b\geq 1$ of the rational
numbers $a/b$ (in lowest terms) that are rotation numbers of simple orbits for
$\lambda$, and let $B=4B_{0}$; if there are no degenerate orbits set
$B_{0}=1$.

  * •

Let $M$ be the constant provided by Lemma 3.9, governing the exceptional pairs
$(\gamma,m)$. By increasing $M$ if necessary, we also assume that $M>B_{0}$.

  * •

Fix $\ell>0$ which is less than the smallest action of a Reeb orbit for
$\lambda$, and also less than the action of all Reeb orbits for $\lambda_{k}$
for every $k$.

  * •

Let $\epsilon_{BM}$ be the constant provided by Lemma 3.7 using $M$ as above,
giving a lower bound on the action difference between orbit sets with low
complexity. By shrinking $\epsilon_{BM}$ if necessary, we can further assume
that $\epsilon_{BM}<\ell$.

  * •

Let $\epsilon^{\prime}=\min\\{\eta,\epsilon_{BM}/2\\}>0$, where $\eta>0$ was
given as an input to Proposition 3.1, see §3.2.2.

  * •

Let $\delta_{1}>0$ be the constant provided by Lemma 3.8 using $\ell$ as
above, controlling the first Chern class of nullhomologous orbit sets.

To define the remaining constants, note that by Proposition 2.10 and our
hypothesis that $c_{1}(\xi)$ is torsion, we can choose a $U$-sequence
$\\{\sigma_{p}\\}_{p\geq p_{0}}$ in $ECH(Y,\xi,0)$ with $p_{0}>0$ such that
each $\sigma_{p}$ has grading $2p$, as defined in equation (2.6). Denote the
corresponding spectral invariant by
$c_{p}=c_{\sigma_{p}}(Y,\lambda)\in\mathbb{R}$.

  * •

By the “Weyl law” of Proposition 2.12, we can fix a constant $\delta_{2}>0$
such that

(3.4) $c_{p}+1\leq\delta_{2}p^{1/2}$

holds for all $p>p_{0}$.

  * •

Fix $q>p_{0}$ to be any integer large enough so that

(3.5) $\displaystyle q^{2/3}$
$\displaystyle>4\delta_{2}q^{1/2}/\ell+6\delta_{1}\delta_{2}q^{1/2}+9\delta_{2}q^{1/2}/\epsilon^{\prime},$
(3.6) $\displaystyle q^{4/5}$
$\displaystyle>4\delta_{2}q^{1/2}/\ell+4\delta_{1}\delta_{2}q^{1/2}+7\left(q^{2/3}+\delta_{2}q^{1/2}/\epsilon^{\prime}\right),$
(3.7) $\displaystyle q^{5/6}$
$\displaystyle>2\delta_{2}q^{1/2}/\ell+4\delta_{1}\delta_{2}q^{1/2}+8\left(\delta_{2}q^{1/2}/\epsilon^{\prime}+q^{4/5}+q^{2/3}\right),$
(3.8) $\displaystyle q-p_{0}$
$\displaystyle>q^{5/6}+q^{4/5}+q^{2/3}+\delta_{2}q^{1/2}/\epsilon^{\prime}.$

We will use these inequalities at the end of the proof in §3.4.1. For now we
note that they can all be satisfied for sufficiently large $q$, where the
demand of sufficiency depends only on $\ell$, $\delta_{1}$, $\delta_{2}$,
$\epsilon^{\prime}$, and $p_{0}$, all of which are $k$-independent.

  * •

Finally, define

(3.9) $T_{q}=\delta_{2}q^{1/2}.$

####  3.2.6. More basic estimates

We will also need two more estimates; these involve the constants from the
previous section.

We have the following lower bound on the energy of holomorphic currents
counted by the $U$-map:

######  Lemma 3.10.

There is a constant $\bar{h}>0$ such that the following holds when $k$ is
sufficiently large: If $\mathcal{C}\in\mathcal{M}^{J_{k}}(\alpha,\beta)$ is a
$U$-curve and $\mathcal{A}(\alpha)<T_{q}$, then

(3.10) $\mathcal{A}(\alpha)-\mathcal{A}(\beta)>\bar{h}.$

###### Proof.

This follows the proof of [8, Lemma 3.1(b)]. The idea is to suppose that there
exists a sequence of $U$-curves
$\mathcal{C}_{i}\in\mathcal{M}^{J_{i}}(\alpha_{i},\beta_{i})$ with
$\mathcal{A}(\alpha_{i})<T_{q}$ and
$\lim_{i\to\infty}(\mathcal{A}(\alpha_{i})-\mathcal{A}(\beta_{i}))=0$. Once
can then use Gromov compactness for currents as proved by Taubes [47, Prop.
3.3] (see [11, Prop. 1.9] for a more general version) to obtain a
contradiction. ∎

######  Lemma 3.11.

If $k$ is sufficiently large, then for any two orbit sets $\alpha$ and $\beta$
for $\lambda_{k}$ with action $\leq T_{q}$ and with $|\alpha|+|\beta|\leq 4$,
if at least one of them has complexity $\leq M$, or if both have exactly one
simple Reeb orbit with multiplicity $>M$ and these simple orbits are the same,
we have either

(3.11) $|\mathcal{A}(\alpha)-\mathcal{A}(\beta)|\leq\bar{h}/2$

or

(3.12) $|\mathcal{A}(\alpha)-\mathcal{A}(\beta)|\geq\epsilon^{\prime}.$

Here the constant $\bar{h}$ is provided by Lemma 3.10. Also recall that
$\epsilon^{\prime}\leq\epsilon_{BM}/2$ where the constant $\epsilon_{BM}$ is
provided by Lemma 3.7.

###### Proof.

Assume to start that $k\geq k_{T_{q}}$ where the constant $k_{T_{q}}$ is given
by Lemma 3.5(c). Recall from §3.2.2 that $\alpha$ and $\beta$ determine orbit
sets $\underline{\alpha}$ and $\underline{\beta}$ for $\lambda$. By Lemma
3.5(c), if $k$ is sufficiently large, then

$|\mathcal{A}(\alpha)-\mathcal{A}(\underline{\alpha})|\leq\bar{h}/4,$

and likewise for $\beta$. Thus if
$\mathcal{A}(\underline{\alpha})=\mathcal{A}(\underline{\beta})$, and if $k$
is sufficiently large, then the inequality (3.11) holds. So we can assume
without loss of generality that
$\mathcal{A}(\underline{\alpha})\neq\mathcal{A}(\underline{\beta})$, and we
want to show that the inequality (3.12) holds if $k$ is sufficiently large.

If $\gamma^{\prime}$ is a simple Reeb orbit of $\lambda$, let
$\mathcal{P}_{k}(\gamma^{\prime})$ denote the set of simple Reeb orbits of
$\lambda$ that are contained in the same component of $N$ as
$\gamma^{\prime}$. Then by Lemma 3.5(c), we have

(3.13) $m(\gamma^{\prime},\underline{\alpha})\leq B_{0}\sum_{\gamma\in
P_{k}(\gamma^{\prime})}m(\gamma,\alpha).$

Since $|\alpha|\leq 4$ by hypothesis, it follows that if $\alpha$ has
complexity $\leq M$, then $\underline{\alpha}$ has complexity $\leq BM$. The
same holds for $\beta$. Thus if $\alpha$ or $\beta$ has complexity $\leq M$,
then Lemma 3.7 applies to give

(3.14)
$|\mathcal{A}(\underline{\alpha})-\mathcal{A}(\underline{\beta})|\geq\epsilon_{BM},$

so that the inequality (3.12) holds if $k$ is sufficiently large.

The remaining possibility is that $\alpha$ and $\beta$ have exactly one simple
Reeb orbit with multiplicity $>M$, and this simple orbit is the same for each,
call it $\gamma$. There is then a simple Reeb orbit $\gamma^{\prime}$ for
$\lambda$ such that $\gamma\in\mathcal{P}_{k}(\gamma^{\prime})$. If
$\gamma^{\prime\prime}$ is a simple Reeb orbit for $\lambda$ with
$\gamma^{\prime\prime}\neq\gamma^{\prime}$, then it follows from (3.13) that
$m(\gamma^{\prime\prime},\underline{\alpha})\leq BM$, and likewise for
$\underline{\beta}$. Thus Lemma 3.7 applies again to give (3.14), so that the
inequality (3.12) holds if $k$ is sufficiently large. ∎

####  3.2.7. A chain of $U$-curves

We now use the $U$-map on ECH to obtain a “chain” of $U$-curves. The following
is loosely related to [9, Lemma 4.4]:

######  Lemma 3.12.

If $k$ is sufficiently large, then there exist:

  * •

ECH generators $\alpha(p_{0}),\ldots,\alpha(q)$ for $\lambda_{k}$, with
$[\alpha(i)]=0\in H_{1}(Y)$ and $\mathcal{A}(\alpha(q))\leq T_{q}$,

  * •

$U$-curves $\mathcal{C}(i)\in\mathcal{M}^{J_{k}}(\alpha(i),\alpha(i-1))$ for
$p_{0}<i\leq q$,

such that:

  * (a)

There are at most $T_{q}/\epsilon^{\prime}$ indices
$i\in\\{p_{0}+1,\ldots,q\\}$ such that

$\mathcal{A}(\alpha(i))-\mathcal{A}(\alpha(i-1))\geq\epsilon^{\prime}.$

  * (b)

We have

(3.15) $\sum^{q}_{i={p_{0}+1}}J_{0}(\mathcal{C}(i))\leq
2(q-p_{0})+2\delta_{1}T_{q}.$

###### Proof.

Since the ECH spectral invariants are $C^{0}$-continuous functions of the
contact form, see §2.5, it follows from the bound (3.4) and equation (3.9)
that if $k$ is sufficiently large then

(3.16) $c_{\sigma_{q}}(Y,\lambda_{k})\leq T_{q}.$

By the definition of the spectral invariant $c_{\sigma_{q}}$, we can find a
cycle $x$ in the ECH chain complex $(ECC(Y,\lambda,0),\partial_{J_{k}})$
representing the class $\sigma_{q}$, such that each ECH generator that is a
summand in $x$ has action $\leq T_{q}$. By the definition of $U$-sequence, we
have $U_{J_{k},y}^{q-p_{0}}x\neq 0$. There is then some summand of $x$, which
we can call $\alpha(q)$, such that $U_{J_{k},y}^{q-p_{0}}x\neq 0$. This means
that there are ECH generators $\alpha(p_{0}),\ldots,\alpha(q-1)$ such that

$\langle U_{J_{k},y}\alpha(i),\alpha(i-1)\rangle\neq 0$

for $p_{0}<i\leq q$. By the definition (2.9) of the chain map $U_{J_{k},y}$,
it follows that for each $i$ with $p_{0}<i\leq q$ there exists a $U$-curve
$\mathcal{C}(i)\in\mathcal{M}^{J_{k}}(\alpha(i),\alpha(i-1))$.

Assertion (a) follows immediately from the fact that
$\mathcal{A}(\alpha(q))\leq T_{q}$ and the nonnegativity of
$\mathcal{A}(\alpha(i))$ and $\mathcal{A}(\alpha(i))-\mathcal{A}(\alpha(i-1))$
for each $i$. It remains to prove assertion (b).

By equation (2.13) and the fact that $I(\mathcal{C}(i))=2$, we have a
telescoping sum

$\begin{split}\sum_{i=p_{0}+1}^{q}J_{0}(\mathcal{C}(i))-2(q-p_{0})&=\sum_{i=p_{0}+1}^{q}\left(J_{0}(\mathcal{C}(i))-I(\mathcal{C}(i))\right)\\\
&=\sum_{i=p_{0}+1}^{q}\left(\left[2c_{\tau}(\alpha(i-1))+{\rm
CZ}_{\tau}^{\operatorname{top}}(\alpha(i-1))\right]-\left[2c_{\tau}(\alpha(i))+{\rm
CZ}_{\tau}^{\operatorname{top}}(\alpha(i))\right]\right)\\\
&=\left[2c_{\tau}(\alpha(p_{0}))+{\rm
CZ}_{\tau}^{\operatorname{top}}(\alpha(p_{0}))\right]-\left[2c_{\tau}(\alpha(q))+{\rm
CZ}_{\tau}^{\operatorname{top}}(\alpha(q)\right].\end{split}$

So to prove part (b), it is enough to show that if $k$ is sufficiently large,
then

(3.17) $\left|2c_{\tau}(\alpha)+{\rm
CZ}_{\tau}^{\operatorname{top}}(\alpha)\right|\leq\delta_{1}T_{q}$

for every ECH generator $\alpha$ for $\lambda_{k}$ with
$\mathcal{A}(\alpha)\leq T_{q}$.

Suppose to start that $k\geq k_{T_{q}}$, where the constant $k_{T_{q}}$ is
provided by Lemma 3.5(c). Recall from §3.2.2 that for each orbit set $\alpha$
for $\lambda_{k}$ with $\mathcal{A}(\alpha)\leq T_{q}$, there is a unique
orbit set $\underline{\alpha}$ for $\lambda$ such that $\alpha$ is homologous
to $\underline{\alpha}$ in $N$. Since the trivialization $\tau$ of $\xi$ over
the Reeb orbits in $\alpha$ and $\underline{\alpha}$ extends to a
trivialization of $\xi$ over all of $N$, it follows from the definition of the
relative first Chern class in §2.1 that

$c_{\tau}(\alpha)=c_{\tau}(\underline{\alpha}).$

By this equation and Lemma 3.8, to prove (3.17), it is enough to show that

$\left|{\rm CZ}_{\tau}^{\operatorname{top}}(\alpha)-{\rm
CZ}_{\tau}^{\operatorname{top}}\left(\underline{\alpha}\right)\right|\leq\frac{\mathcal{A}(\alpha)}{\ell}.$

By our choice of $\ell$ in §3.2.5, there are at most
$\mathcal{A}(\alpha)/\ell$ Reeb orbits in $\alpha$. So to complete the proof,
it is enough to show that if $k\geq k_{T_{q}}$ is sufficiently large, then for
every (not necessarily simple) Reeb orbit $\gamma$ of $\lambda_{k}$ with
$\mathcal{A}(\gamma)\leq T_{q}$, if $\underline{\gamma}$ is the corresponding
Reeb orbit of $\lambda$ (in the notation of Lemma 3.5(c), $\underline{\gamma}$
is either $\gamma^{\prime}$ or $(\gamma^{\prime})^{b}$), then

(3.18) $\left|{\rm CZ}_{\tau}(\gamma)-{\rm
CZ}_{\tau}\left(\underline{\gamma}\right)\right|\leq 1.$

By Lemma 3.5(c) and equation (2.1), the bound (3.18) holds when $k$ is
sufficiently large. ∎

###  3.3. The score

To analyze the chain of $U$-curves produced in §3.2.7, we now define a “score”
associated to each $U$-curve and analyze it in detail.

####  3.3.1. Definition of the score

######  Definition 3.13.

Let $\alpha$ be an orbit set for $\lambda_{k}$, and let $\gamma$ be a simple
Reeb orbit of $\lambda_{k}$.

  * •

We say that $\gamma$ is a special component of $\alpha$ if
$m(\gamma,\alpha)>1$ and $1\notin p_{\gamma}^{+}(m(\gamma,\alpha))$. Define
$s(\gamma,\alpha)$ to be $1$ if $\gamma$ is a special component of $\alpha$,
and $0$ otherwise. Denote the number of special components of $\alpha$ by

$s(\alpha)=\sum_{\gamma\in\mathcal{P}_{k}}s(\gamma,\alpha).$

  * •

We say that $\gamma$ is a $p^{+}$-component of $\alpha$ if
$p_{\gamma}^{+}(m(\gamma,\alpha))=(m(\gamma,\alpha))$ and
$m(\gamma,\alpha)\geq M$, where $M$ is the constant provided by Lemma 3.9.
Define $p^{+}(\gamma,\alpha)$ to be $1$ if $\gamma$ is a $p^{+}$-component of
$\alpha$, and $0$ otherwise. Denote the number of $p^{+}$-components of
$\alpha$ by

$p^{+}(\alpha)=\sum_{\gamma\in\mathcal{P}_{k}}p^{+}(\gamma,\alpha).$

  * •

Similarly, we say that $\gamma$ is a $p^{-}$-component of $\alpha$ if
$p_{\gamma}^{-}(m(\gamma,\alpha))=(m(\gamma,\alpha))$ and
$m(\gamma,\alpha)\geq M$. Define $p^{-}(\gamma,\alpha)$ to be $1$ if $\gamma$
is a $p^{-}$-component of $\alpha$, and $0$ otherwise. Denote the number of
$p^{-}$-components of $\alpha$ by

$p^{-}(\alpha)=\sum_{\gamma\in\mathcal{P}_{k}}p^{-}(\gamma,\alpha).$

######  Definition 3.14.

If $\alpha$ is an orbit set for $\lambda_{k}$, its orbit score is the quantity

$S(\alpha):=p^{+}(\alpha)+s(\alpha)-p^{-}(\alpha).$

######  Definition 3.15.

Let $\mathcal{C}\in\mathcal{M}^{J_{k}}(\alpha,\beta)$ be a $U$-curve. The
orbit score of $\mathcal{C}$ is

$S(\mathcal{C})=S(\alpha)-S(\beta).$

The total score of $\mathcal{C}$ (also called the “score” when there is no
danger of confusion) is

$T(\mathcal{C})=3y(\mathcal{C})+S(\mathcal{C})$

where we define

$y(\mathcal{C})=J_{0}(\mathcal{C})-2.$

######  Definition 3.16.

If $\mathcal{C}$ is a $U$-curve as above, and if $\gamma$ is a simple Reeb
orbit of $\lambda_{k}$, we define the contribution to the orbit score of
$\mathcal{C}$ associated to $\gamma$ by

(3.19)
$S_{\gamma}(\mathcal{C})=\left(p^{+}(\gamma,\alpha)+s(\gamma,\alpha)+p^{-}(\gamma,\beta)\right)-\left(p^{+}(\gamma,\beta)+s(\gamma,\beta)+p^{-}(\gamma,\alpha)\right).$

We define the contribution to the total score of $\mathcal{C}$ associated to
$\gamma$ by

(3.20)
$T_{\gamma}(\mathcal{C})=S_{\gamma}(\mathcal{C})+3e_{\gamma}(\mathcal{C}),$

where $e_{\gamma}(\mathcal{C})$ was defined in §2.7.

######  Remark 3.17.

It follows from equation (2.15) and the above definitions that if we write
$\mathcal{C}=\mathcal{C}_{0}+C_{1}$ as in Proposition 2.6, then

(3.21)
$T(\mathcal{C})=6g(C_{1})-12+\sum_{\gamma\in\mathcal{P}_{k}}T_{\gamma}(\mathcal{C}).$

Moreover, in the sum on the right, the contribution $T_{\gamma}(\mathcal{C})$
is nonzero only if $C_{1}$ has an end asymptotic to a cover of $\gamma$.

####  3.3.2. How orbits contribute to the score

We now collect some lemmas about how orbits contribute to the score of a
$U$-curve.

######  Lemma 3.18.

If $k$ is sufficiently large then the following holds. Let
$\mathcal{C}=\mathcal{C}_{0}+C_{1}\in\mathcal{M}^{J_{k}}(\alpha,\beta)$ be a
$U$-curve such that $\mathcal{A}(\alpha)\leq T_{q}$. Let
$\gamma\in\mathcal{P}_{k}$ and suppose that $C_{1}$ has at least one end
asymptotic to a cover of $\gamma$. Let $m_{0}$ denote the multiplicity of the
trivial cylinder $\mathbb{R}\times\gamma$ in $\mathcal{C}_{0}$. Then:

  * (a)

$T_{\gamma}(\mathcal{C})\geq 3$.

  * (b)

If $T_{\gamma}(\mathcal{C})=3$, then $m_{0}=0$, the curve $C_{1}$ has a unique
end at a cover of $\gamma$, and if this end is positive then the covering
multiplicity must be one.

###### Proof.

By equation (3.19) we have $S_{\gamma}(\mathcal{C})\geq-3$. If
$e_{\gamma}(\mathcal{C})\geq 3$ then it follows from equation (3.20) that
$T_{\gamma}(\mathcal{C})\geq 6$. It therefore remains to consider the cases
where $e_{\gamma}(\mathcal{C})\in\\{1,2\\}$.

Case 1: Suppose that $e_{\gamma}(\mathcal{C})=1$. In this case, it follows
from the definition of $e_{\gamma}$ in §2.7 that $m_{0}=0$, and the curve
$C_{1}$ has just one end at a cover of $\gamma$.

Subcase 1a: Suppose that the end of $C_{1}$ at a cover of $\gamma$ is a
positive end. Then $p^{+}(\gamma,\beta)=s(\gamma,\beta)=0$.

If $m(\gamma,\alpha)=1$, then $p^{-}(\gamma,\alpha)=0$ (since $M>1$), so
$S_{\gamma}(\mathcal{C})\geq 0$, so $T_{\gamma}(\mathcal{C})\geq 3$, and the
conditions for equality in (b) hold.

If $m(\gamma,\alpha)>1$, then it follows from Proposition 2.16 that
$p_{\gamma}^{+}(m(\gamma,\alpha))=(m(\gamma,\alpha))$, so
$s(\gamma,\alpha)=1$. Since $\alpha$ is an ECH generator, $\gamma$ has
irrational rotational number, so it follows from Lemma 2.18(b) that
$p^{-}_{\gamma}(m(\gamma,\alpha))\neq(m(\gamma,\alpha))$. This implies that
$p^{-}(\gamma,\alpha)=0$. We conclude that $S_{\gamma}(\mathcal{C})>0$, so
$T_{\gamma}(\mathcal{C})>3$.

Subcase 1b: Suppose that the end of $C_{1}$ at a cover of $\gamma$ is a
negative end.

Then $p^{-}(\gamma,\alpha)=0$. Similarly to the above paragraph, it follows
from Proposition 2.16 that $p_{\gamma}^{-}(m(\gamma,\beta))=(m(\gamma,\beta))$
and then from Lemma 2.18(b) that $p^{+}(\gamma,\beta)=0$. By Lemma 2.18(a), we
have $1\in p_{\gamma}^{+}(m(\gamma,\beta))$, so $s(\gamma,\beta)=0$. Thus
$S_{\gamma}(\mathcal{C})\geq 0$, so $T_{\gamma}(\mathcal{C})\geq 3$, and the
conditions for equality in (b) hold.

Case 2: Suppose that $e_{\gamma}(\mathcal{C})=2$. It will suffice to show that
$S_{\gamma}(\mathcal{C})\geq-2$, so that $T_{\gamma}(C)>3$.

Subcase 2a: Suppose that $m_{0}=0$. Then by the definition of $e_{\gamma}$,
the curve $C_{1}$ has a positive end and a negative end asymptotic to covers
of $\gamma$. Then $s(\gamma,\beta)=0$ by the same argument as in Subcase 1b,
so $S_{\gamma}(\mathcal{C})\geq-2$.

Subcase 2b: Suppose that $m_{0}>0$. Then $C_{1}$ has a unique end asymptotic
to a cover of $\gamma$. To prove that $S_{\gamma}(\mathcal{C})\geq-2$, suppose
to get a contradiction that $S_{\gamma}(\mathcal{C})=-3$. Then $\gamma$ is
both a $p^{-}$-component of $\alpha$ and a $p^{+}$-component of $\beta$.

Consider the case where the end of $C_{1}$ asymptotic to a cover of $\gamma$
is a positive end, and let $d$ denote the covering multiplicity. Then
$m(\gamma,\beta)=m_{0}$, and $m(\gamma,\alpha)=m(\gamma,\beta)+d$. By
Proposition 2.16,

$\begin{split}p_{\gamma}^{+}(m(\gamma,\alpha))&=p_{\gamma}^{+}(m(\gamma,\beta))\cup(d)\\\
&=(m(\gamma,\beta),d)\end{split}$

since $\gamma$ is a $p^{+}$-component of $\beta$. In particular
$|p_{\gamma}^{+}(m(\gamma,\alpha))|=2$.

Since $\gamma$ is a $p^{-}$-component of $\alpha$, we also have
$|p_{\gamma}^{-}(m(\gamma,\alpha))|=1$. We conclude that the pair
$(\gamma,m(\gamma,\alpha))$ is exceptional. Since $m(\gamma,\alpha)\geq M$ by
the definition of “$p^{+}$-component”, this contradicts Lemma 3.9 if $k\geq
k_{T_{q}}^{\prime}$.

The case when the end of $C_{1}$ asymptotic to a cover of $\gamma$ is a
negative end is handled by a symmetric argument. ∎

The next lemma gives a refinement of the previous one when we assume in
addition that $C_{1}$ has a “high multiplicity” positive end.

######  Definition 3.19.

A positive or negative end of a $J_{k}$-holomorphic curve asymptotic to the
$d$-fold cover of an orbit $\gamma\in\mathcal{P}_{k}$ is high multiplicity if
$d\geq M$.

######  Lemma 3.20.

If $k$ is sufficiently large then the following holds. Let
$\mathcal{C}=\mathcal{C}_{0}+C_{1}\in\mathcal{M}^{J_{k}}(\alpha,\beta)$ be a
$U$-curve such that $\mathcal{A}(\alpha)\leq T_{q}$. Let
$\gamma\in\mathcal{P}_{k}$ and suppose that $C_{1}$ has at least one high
multiplicity positive end asymptotic to a cover of $\gamma$. Let $m_{0}$
denote the multiplicity of the trivial cylinder $\mathbb{R}\times\gamma$ in
$\mathcal{C}_{0}$. Then:

  * (a)

If $C_{1}$ has a unique end asymptotic to a cover of $\gamma$, then
$T_{\gamma}(\mathcal{C})\geq 5$, with equality only if $m_{0}\neq 1$.

  * (b)

If $C_{1}$ has at least two ends asymptotic to covers of $\gamma$, then
$T_{\gamma}(\mathcal{C})\geq 8$, with equality only if $C_{1}$ has exactly one
positive and one negative end asymptotic to covers of $\gamma$ and $m_{0}=0$.

###### Proof.

(a) Suppose that $C_{1}$ has a unique end asymptotic to a cover of $\gamma$.

Case 0: Suppose that $m_{0}=0$. We need to show that
$T_{\gamma}(\mathcal{C})\geq 5$.

In this case $e_{\gamma}(\mathcal{C})=1$, so we need to show that
$S_{\gamma}(\mathcal{C})\geq 2$. From Proposition 2.16 and Lemma 2.18(b), we
learn that $\gamma$ is a $p^{+}$-component and a special component but not a
$p^{-}$-component of $\alpha$. Hence $S_{\gamma}(\mathcal{C})=2$.

Case 1: Suppose that $m_{0}=1$. We need to show that
$T_{\gamma}(\mathcal{C})\geq 6$.

In this case $e_{\gamma}(\mathcal{C})=2$, so we need to show that
$S_{\gamma}(\mathcal{C})\geq 0$. Since $C_{1}$ has no negative ends asymptotic
to covers of $\gamma$, it follows that
$p^{+}(\gamma,\beta)=s(\gamma,\beta)=0$. In addition, if $k\geq
k_{T_{q}}^{\prime\prime}$, then $p^{-}(\gamma,\alpha)=0$, because otherwise
the pair $(\gamma,m(\gamma,\alpha))$ would be exceptional by Proposition 2.16,
and this would contradict Lemma 3.9 because $m(\gamma,\alpha)\geq M+1$. It
then follows from equation (3.19) that $S_{\gamma}(\mathcal{C})\geq 0$.

Case 2: Suppose that $m_{0}\geq 2$. We need to show that
$T_{\gamma}(\mathcal{C})\geq 5$.

In this case $e_{\gamma}(\mathcal{C})=2$ as above, so we need to show that
$S_{\gamma}(\mathcal{C})\geq-1$. By Proposition 2.16 and Remark 2.17, we have
$1\in p_{\gamma}^{+}(m(\gamma,\alpha))$ if and only if $1\in
p_{\gamma}^{+}(m_{0})$. Since $m_{0}=m(\gamma,\beta)$, it follows that
$s(\gamma,\alpha)-s(\gamma,\beta)=0$.

To finish showing that $S_{\gamma}(\mathcal{C})\geq-1$, we observe that if
$k\geq k_{T_{q}}^{\prime}$ then $\gamma$ cannot be both a $p^{-}$-component of
$\alpha$ and a $p^{+}$-component of $\beta$. Otherwise the pair
$(\gamma,m(\gamma,\alpha))$ would be exceptional by Proposition 2.16,
contradicting Lemma 3.9 since $m(\gamma,\alpha)\geq M+2$.

(b) Suppose that $C_{1}$ has at least two ends asymptotic to covers of
$\gamma$.

Note first that if $m_{0}>0$, or if $C_{1}$ has at least three ends asymptotic
to covers of $\gamma$, then $e_{\gamma}(\mathcal{C})\geq 4$, and so
$T_{\gamma}(\mathcal{C})\geq 9$. Thus we can assume that $m_{0}=0$ and that
$C_{1}$ has exactly two ends asymptotic to covers of $\gamma$.

Case 1: Suppose that $C_{1}$ has two positive ends asymptotic to covers of
$\gamma$. We need to show that $T_{\gamma}(\mathcal{C})\geq 9$.

In this case $e_{\gamma}(\mathcal{C})=3$, so we need to show that
$S_{\gamma}(\mathcal{C})\geq 0$. Since $m(\gamma,\beta)=0$, it is enough to
show that $\gamma$ is not a $p^{-}$-component of $\alpha$. But if $\gamma$
were a $p^{-}$-component of $\alpha$, then the pair
$(\gamma,m(\gamma,\alpha))$ would be exceptional by Proposition 2.16,
contradicting Lemma 3.9 if $k\geq k_{T_{q}}^{\prime}$.

Case 2: Suppose that $C_{1}$ has a positive end and a negative end asymptotic
to covers of $\gamma$. We need to show that $T_{\gamma}(\mathcal{C})\geq 8$.

In this case $e_{\gamma}(\mathcal{C})=2$, so we need to show that
$S_{\gamma}(\mathcal{C})\geq 2$. By Proposition 2.16,
$p_{\gamma}^{+}(m(\gamma,\alpha))=(m(\gamma,\alpha))$, so $\gamma$ is a
$p^{+}$-component and special component of $\alpha$. By Lemma 2.18(b),
$\gamma$ cannot be a $p^{-}$-component of $\alpha$. By Proposition 2.16 again,
$p_{\gamma}^{-}(m(\gamma,\beta))=(m(\gamma,\beta))$. So by Lemma 2.18(b),
$\gamma$ cannot be a $p^{+}$-component of $\beta$, and by Lemma 2.18(a),
$\gamma$ cannot be a special component of $\beta$. Thus
$S_{\gamma}(\mathcal{C})\geq 2$ by equation (3.19). ∎

The next lemma gives a similar refinement of Lemma 3.18 when we assume in
addition that $C_{1}$ has a high multiplicity negative end.

######  Lemma 3.21.

If $k$ is sufficiently large then the following holds. Let
$\mathcal{C}=\mathcal{C}_{0}+C_{1}\in\mathcal{M}^{J_{k}}(\alpha,\beta)$ be a
$U$-curve such that $\mathcal{A}(\alpha)\leq T_{q}$. Let
$\gamma\in\mathcal{P}_{k}$ and suppose that $C_{1}$ has at least one high
multiplicity negative end asymptotic to a cover of $\gamma$. Let $m_{0}$
denote the multiplicity of the trivial cylinder $\mathbb{R}\times\gamma$ in
$\mathcal{C}_{0}$. Then:

  * (a)

If $C_{1}$ has a unique end asymptotic to a cover of $\gamma$, then
$T_{\gamma}(\mathcal{C})\geq 4$, with equality only if $m_{0}=0$.

  * (b)

If $C_{1}$ has at least two ends asymptotic to covers of $\gamma$, then
$T_{\gamma}(\mathcal{C})\geq 7$, with equality only if $C_{1}$ has exactly one
positive and one negative end asymptotic to covers of $\gamma$, the covering
multiplicity for the positive end is one, and $m_{0}=0$.

###### Proof.

(a) Suppose that $C_{1}$ has exactly one end asymptotic to a cover of
$\gamma$.

Case 0: Suppose that $m_{0}=0$. We need to show that
$T_{\gamma}(\mathcal{C})\geq 4$.

In this case $e_{\gamma}(\mathcal{C})=1$, so we need to show that
$S_{\gamma}(\mathcal{C})\geq 1$. By Proposition 2.16, we have
$p_{\gamma}^{-}(m(\gamma,\beta))=(m(\gamma,\beta))$, so
$p^{-}(\gamma,\beta)=1$. By Lemma 2.18(b), we have $p^{+}(\gamma,\beta)=0$. By
Lemma 2.18(a), we have $s(\gamma,\beta)=0$. Hence $S_{\gamma}(\mathcal{C})=1$.

Case 1: Suppose that $m_{0}=1$. We need to show that
$T_{\gamma}(\mathcal{C})\geq 5$.

In this case $e_{\gamma}(\mathcal{C})=2$, so we need to show that
$S_{\gamma}(\mathcal{C})\geq-1$. We have $p^{-}(\gamma,\alpha)=0$ by
definition, because $m(\gamma,\alpha)=1$. By Proposition 2.16, we have
$p_{\gamma}^{-}(m(\gamma,\beta))=(m(\gamma,\beta)-1,1)$. If $k\geq
k_{T_{q}}^{\prime\prime}$ then it follows that $p^{+}(\gamma,\beta)=0$, or
else the pair $(\beta,m(\gamma,\beta))$ would be exceptional, contradicting
Lemma 3.9. Hence there is only one possible negative term in equation (3.19).

Case 2: Suppose that $m_{0}>1$. Again we need to show that
$T_{\gamma}(\mathcal{C})\geq 5$. This is proved by a symmetric argument to
Case 2 of the proof of Lemma 3.20(a).

(b) Suppose that $C_{1}$ has at least two ends asymptotic to covers of
$\gamma$. As in the proof of Lemma 3.20(b), we can assume that $m_{0}=0$ and
that $C_{1}$ has exactly two ends asymptotic to covers of $\gamma$.

Case 1: Suppose that $C_{1}$ has two negative ends asymptotic to covers of
$\gamma$. We need to show that $T_{\gamma}(\mathcal{C})\geq 8$.

In this case $e_{\gamma}(\mathcal{C})=3$, so we need to show that
$S_{\gamma}(\mathcal{C})\geq-1$. This holds if $k\geq
k_{T_{q}}^{\prime\prime}$, because $p^{+}(\gamma,\beta)=0$, because otherwise
the pair $(\gamma,m(\gamma,\beta))$ would be exceptional, contradicting Lemma
3.9.

Case 2: Suppose that $C_{1}$ has exactly one positive and one negative end
asymptotic to covers of $\gamma$. We need to show that
$T_{\gamma}(\mathcal{C})\geq 7$, with equality only if $m(\gamma,\alpha)=1$.

In this case $e_{\gamma}(\mathcal{C})=2$, so we need to show that
$S_{\gamma}(\mathcal{C})\geq 1$, with equality only if $m(\gamma,\alpha)=1$.
By Proposition 2.16, $p_{\gamma}^{+}(m(\gamma,\alpha))=(m(\gamma,\alpha))$ and
$p_{\gamma}^{-}(m(\gamma,\beta))=(m(\gamma,\beta))$. Thus $\gamma$ is a
$p^{-}$-component of $\beta$, and $\gamma$ is a special component of $\alpha$
if $m(\gamma,\alpha)>1$. By Lemma 2.18(b), $\gamma$ is not a $p^{+}$-component
of $\beta$ or a special component of $\beta$ or a $p^{-}$-component of
$\alpha$. We are now done by equation (3.19). ∎

####  3.3.3. Nonnegativity of the score

The following proposition is our main interest in the score: it asserts that
low energy curves in our $U$-sequence have nonnegative score, and in fact
positive score except in a few cases that can be classified.

######  Lemma 3.22.

If $k$ is sufficiently large then the following holds. Let
$\mathcal{C}=\mathcal{C}_{0}+C_{1}\in\mathcal{M}^{J_{k}}(\alpha,\beta)$ be a
$U$-curve such that $\mathcal{A}(\alpha)\leq T_{q}$. Assume that:

  * •

$\mathcal{A}(\alpha)-\mathcal{A}(\beta)<\epsilon^{\prime}$, where the constant
$\epsilon^{\prime}$ was specified in §3.2.5.

  * •

$C_{1}$ is not a cylinder.

Then:

  * (a)

The total score satisfies $T(\mathcal{C})\geq 0$.

  * (b)

If $T(\mathcal{C})=0$, then:

    * (1)

$C_{1}$ has genus zero and exactly three ends.

    * (2)

There are unique, distinct simple Reeb orbits
$\gamma_{1},\gamma_{2}\in\mathcal{P}_{k}$ such that $C_{1}$ has a high
multiplicity positive end asymptotic to a cover of $\gamma_{1}$ and a high
multiplicity negative end asymptotic to a cover of $\gamma_{2}$.

    * (3)

Either $C_{1}$ has an additional positive end asymptotic to a simple Reeb
orbit distinct from $\gamma_{1}$, or $C_{1}$ has an additional negative end
asymptotic to a cover of a simple Reeb orbit distinct from $\gamma_{2}$.

    * (4)

If a trivial cylinder $\mathbb{R}\times\gamma$ appears in $\mathcal{C}_{0}$,
and if $C_{1}$ has an end asymptotic to a cover of $\gamma$, then
$\gamma=\gamma_{1}$, and the multiplicity of $\mathbb{R}\times\gamma_{1}$ in
$\mathcal{C}_{0}$ is at least $2$.

    * (5)

$J_{0}(\mathcal{C})=2$ if $\mathbb{R}\times\gamma_{1}$ appears in
$\mathcal{C}_{0}$, and $J_{0}(\mathcal{C})=1$ otherwise.

###### Proof.

Let $\alpha(1)$ denote the positive orbit set of $C_{1}$, and let $\beta(1)$
denote the negative orbit set of $C_{1}$. Note that
$C_{1}\in\mathcal{M}^{J_{k}}(\alpha(1),\beta(1))$ is also a $U$-curve.

Suppose first that $C_{1}$ has ends asymptotic to covers of at least $5$
distinct simple Reeb orbits of $\lambda_{k}$. Then it follows from Remark 3.17
and Lemma 3.18(a) that $T(\mathcal{C})\geq 3$, so the proposition is true in
this case.

It remains to consider the case where $C_{1}$ has ends asymptotic to covers of
at most $4$ distinct simple Reeb orbits, i.e. $|\alpha(1)|+|\beta(1)|\leq 4$.
By Lemma 3.10, if $k$ is sufficiently large then the inequality (3.10) holds.
Assume that $k$ is sufficiently large in this sense and also sufficiently
large so that Lemma 3.11 applies. Then by Lemma 3.11, there are distinct
simple Reeb orbits $\gamma_{1},\gamma_{2}\in\mathcal{P}_{k}$ such that $C_{1}$
has a high multiplicity positive end asymptotic to a cover of $\gamma_{1}$ and
a high multiplicity negative end asymptotic to a cover of $\gamma_{2}$. We now
consider two cases, depending on whether or not $C_{1}$ has ends asymptotic to
covers of any additional simple Reeb orbits.

Case 1: Suppose that $C_{1}$ has an end asymptotic to a cover of a simple Reeb
orbit $\gamma_{3}$ distinct from $\gamma_{1},\gamma_{2}$. Let $m_{i}$ denote
the multiplicity of the trivial cylinder $\mathbb{R}\times\gamma_{i}$ in
$\mathcal{C}_{0}$ for $i=1,2,3$. Then by Remark 3.17 and Lemmas 3.18(a), 3.20,
and 3.21, we have

(3.22) $\begin{split}T(\mathcal{C})&\geq
6g(C_{1})-12+T_{\gamma_{1}}(\mathcal{C})+T_{\gamma_{2}}(\mathcal{C})+T_{\gamma_{3}}(\mathcal{C})\\\
&\geq 6g(C_{1})-12+5+4+3\\\ &\geq 0.\end{split}$

This proves (a). To prove (b), suppose that equality holds in (3.22). Then
$g(C_{1})=0$, the curve $C_{1}$ does not have ends asymptotic to covers of any
orbits other than $\gamma_{1}$, $\gamma_{2}$, or $\gamma_{3}$, and we have
$T_{\gamma_{1}}(\mathcal{C})=5$, $T_{\gamma_{2}}(\mathcal{C})=4$, and
$T_{\gamma_{3}}(\mathcal{C})=3$. In particular, the end asymptotic to a cover
of $\gamma_{3}$ is not high multiplicity. By Lemma 3.20, $C_{1}$ has no other
ends asymptotic to covers of $\gamma_{1}$, and $m_{1}\neq 1$. By Lemma 3.21,
$C_{1}$ has no other ends asymptotic to covers of $\gamma_{2}$, and $m_{2}=0$.
By Lemma 3.18, $m_{3}=0$, and $C_{1}$ has only one end asymptotic to a cover
of $\gamma_{3}$. Moreover, if this end is a positive end, then the covering
multiplicity is $1$. Thus assertions (1)–(4) in the proposition hold.
Assertion (5) follows from equation (2.15).

Case 2: Suppose that $C_{1}$ does not have ends asymptotic to covers of any
simple orbits other than $\gamma_{1}$ and $\gamma_{2}$. In this case,
similarly to (3.22), we have

$T(\mathcal{C})\geq 6g(C_{1})-3.$

We can assume that $C_{1}$ has genus $0$, or else $T(\mathcal{C})>0$ and we
are done. Thus by Remark 3.17 we have

(3.23)
$T(\mathcal{C})=-12+T_{\gamma_{1}}(\mathcal{C})+T_{\gamma_{2}}(\mathcal{C}).$

Since we assumed that $C_{1}$ is not a cylinder, it must have at least two
ends asymptotic to covers of $\gamma_{1}$ or at least two ends asymptotic to
covers of $\gamma_{2}$.

Suppose first that $C_{1}$ has at least two ends asymptotic to covers of
$\gamma_{1}$. Then by equation (3.22) and Lemmas 3.20 and 3.21, we have
$T(\mathcal{C})\geq 0$, with equality only if $T_{\gamma_{1}}(\mathcal{C})=8$
and $T_{\gamma_{2}}(\mathcal{C})=4$. If the latter equalities hold, then
$C_{1}$ has exactly one positive end and one negative end asymptotic to covers
of $\gamma_{1}$, the curve $C_{1}$ has only one end asymptotic to a cover of
$\gamma_{2}$, and $\mathcal{C}_{0}$ does not include the trivial cylinders
$\mathbb{R}\times\gamma_{1}$ or $\mathbb{R}\times\gamma_{2}$. Thus
$J_{0}(\mathcal{C})=1$ by equation (2.15).

Finally suppose that $C_{1}$ has at least two ends asymptotic to covers of
$\gamma_{2}$. Then by equation (3.22) and Lemmas 3.20 and 3.21, we have
$T(\mathcal{C})\geq 0$, with equality only if $T_{\gamma_{1}}(\mathcal{C})=5$
and $T_{\gamma_{2}}(\mathcal{C})=7$. If the latter equalities hold, then
$C_{1}$ has only one end asymptotic to a cover of $\gamma_{1}$, and $m_{1}\neq
1$; moreover $C_{1}$ has exactly one positive and one negative end asymptotic
to covers of $\gamma_{2}$, the covering multiplicity for the positive end is
$1$, and $m_{2}=0$. Thus assertions (1)–(4) hold, and assertion (5) follows
from (2.15) as before. ∎

###  3.4. Putting it all together

We can now complete the proof of Proposition 3.1.

####  3.4.1. Finding a low energy cylinder

We first prove a lemma which provides the cylinders claimed by Proposition
3.1. Assume that $k$ is sufficiently large so that Lemma 3.12 is applicable,
and let $\mathcal{C}(i)\in\mathcal{M}^{J_{k}}(\alpha(i),\alpha(i-1))$ for
$p_{0}<i\leq q$ be a chain of $U$-curves provided by Lemma 3.12. Below we say
that $\mathcal{C}(i)$ is low energy if

$\mathcal{A}(\alpha(i))-\mathcal{A}(\alpha(i-1))<\epsilon^{\prime};$

otherwise we say that $\mathcal{C}(i)$ is high energy. Also, if we write a
$U$-curve $\mathcal{C}$ in the usual manner as $\mathcal{C}_{0}+C_{1}$, then
we refer to $C_{1}$ as the nontrivial component of $\mathcal{C}$.

######  Lemma 3.23.

If $k$ is sufficiently large, then there exists $i\in\\{p_{0}+1,\ldots,q\\}$
such that the $U$-curve $\mathcal{C}(i)$ is low energy, and the nontrivial
component of $\mathcal{C}(i)$ is a cylinder.

###### Proof.

Suppose to get a contradiction that there is no $i$ with $p_{0}<i\leq q$ such
that $\mathcal{C}(i)$ is low energy and the nontrivial component of
$\mathcal{C}(i)$ is a cylinder. By this hypothesis and Lemma 3.22, we can
decompose

(3.24) $\\{p_{0}+1,\ldots,q\\}=G_{0}\sqcup G_{1}\sqcup G_{2}\sqcup G_{3}$

where

$\begin{split}G_{0}&=\\{i\mid\text{$\mathcal{C}(i)$ is high energy}\\},\\\
G_{1}&=\\{i\mid\text{$\mathcal{C}(i)$ is low energy, $T(\mathcal{C}(i))=0$,
$J_{0}(\mathcal{C}(i))=1$}\\},\\\ G_{2}&=\\{i\mid\text{$\mathcal{C}(i)$ is low
energy, $T(\mathcal{C}(i))=0$, $J_{0}(\mathcal{C}(i))=2$}\\},\\\
G_{3}&=\\{i\mid\text{$\mathcal{C}(i)$ is low energy,
$T(\mathcal{C}(i))>0$}\\}.\end{split}$

The strategy will be to put bounds on the cardinalities of the above four
subsets, and to contradict (3.24) by showing that the total cardinality is too
small. We proceed in five steps.

Step 1. We first put bounds on $|G_{0}|$.

Note that since $\mathcal{A}(\alpha(q))\leq\delta_{2}q^{1/2}$ by (3.16), we
have

(3.25) $|G_{0}|\leq\delta_{2}q^{1/2}/\epsilon^{\prime}\,.$

Note also that for each $i$ we have the coarse bound

(3.26) $T(\mathcal{C}(i))\geq-9,$

by Remark 3.17 and Lemma 3.18. It follows from (3.26) and Lemma 3.12(a) that

(3.27) $\sum_{i\in
G_{0}}T(\mathcal{C}(i))\geq-9\delta_{2}q^{1/2}/\epsilon^{\prime}\,.$

Step 2. We now show that $|G_{3}|<q^{2/3}$.

To start, observe that

$\displaystyle\sum^{q}_{i=p_{0}+1}T(\mathcal{C}(i))$
$\displaystyle=S(\alpha(q))-S(\alpha(p_{0}))+3\sum^{q}_{i=p_{0}+1}y(\mathcal{C}(i))$
$\displaystyle=[S(\alpha(q))-S(\alpha(p_{0}))]+3\left(-2(q-p_{0})+\sum^{q}_{i=p_{0}+1}J_{0}(\mathcal{C}(i))\right)$
$\displaystyle\leq[S(\alpha(q))-S(\alpha(p_{0}))]+6\delta_{1}\delta_{2}q^{1/2}$
(3.28) $\displaystyle\leq
4\delta_{2}q^{1/2}/\ell+6\delta_{1}\delta_{2}q^{1/2}.$

Here the first line uses a telescoping sum, the second line uses the fact that
$y=-2+J_{0}$, and the the third line uses the inequality (3.15). The fourth
line holds because if $\alpha=\alpha(q)$ or $\alpha=\alpha(p_{0})$ then we
have $|S(\alpha))|\leq 2|\alpha|$ by definition, and
$\mathcal{A}(\alpha)\geq|\alpha|\ell$; on the other hand,
$\mathcal{A}(\alpha)\leq\delta_{2}q^{1/2}$ by (3.16).

Now by definition, if $i\in G_{3}$ then $T(\mathcal{C}(i))\geq 1$, and if
$i\in G_{1}\sqcup G_{2}$ then $T(\mathcal{C}(i))=0$. Thus

$\begin{split}|G_{3}|&\leq\sum_{i\in G_{1}\sqcup G_{2}\sqcup
G_{3}}T(\mathcal{C}(i))\\\ &=\sum_{i=p_{0}+1}^{q}T(\mathcal{C}(i))-\sum_{i\in
G_{0}}T(\mathcal{C}(i))\\\ &\leq
4\delta_{2}q^{1/2}/\ell+6\delta_{1}\delta_{2}q^{1/2}+9\delta_{2}q^{1/2}/\epsilon^{\prime}\\\
&<q^{2/3}\end{split}$

Here the second to last line uses (3.28) and (3.27), while the last line uses
(3.5).

Step 3. We now show that $|G_{1}|<q^{4/5}$.

To do so, we will use a slight variant of the total score. If $\mathcal{C}$ is
a $U$-curve, define

$T^{\prime}(\mathcal{C})=2y(\mathcal{C})+S(\mathcal{C})=T(\mathcal{C})-y(\mathcal{C}).$

If $i\in G_{1}$, then $T^{\prime}(\mathcal{C}(i))=1$. The reason is that by
the definition of $G_{1}$, we have $T(\mathcal{C}(i))=0$ and
$y(\mathcal{C}(i))=J_{0}(\mathcal{C}(i))-2=-1$.

To bound $T^{\prime}$ of other $U$-curves, if $\gamma$ is a simple Reeb orbit
of $\lambda_{k}$, and if $\mathcal{C}$ is a $U$-curve, then analogously to
(3.20), define

$T^{\prime}_{\gamma}(\mathcal{C})=S_{\gamma}(\mathcal{C})+2e_{\gamma}(\mathcal{C}).$

As in Remark 3.17, we have

(3.29)
$T^{\prime}(\mathcal{C})=4g(C_{1})-8+\sum_{\gamma\in\mathcal{P}_{k}}T^{\prime}_{\gamma}(\mathcal{C}).$

Moreover in the sum on the right, the contribution
$T^{\prime}_{\gamma}(\mathcal{C})$ is nonzero only if $C_{1}$ has an end
asymptotic to a cover of $\gamma$. We now have the following variant of Lemma
3.18(a):

######  Lemma 3.24.

Let $\mathcal{C}=\mathcal{C}_{0}+C_{1}\in\mathcal{M}^{J_{k}}(\alpha,\beta)$ be
a $U$-curve, and let $\gamma\in\mathcal{P}_{k}$. Suppose that $C_{1}$ has at
least one end asymptotic to a cover of $\gamma$. Then
$T_{\gamma}^{\prime}(\mathcal{C})\geq 1$.

###### Proof.

By hypothesis, $e_{\gamma}(\mathcal{C})>0$. If $e_{\gamma}(\mathcal{C})\geq
2$, then the lemma is true automatically since $S_{\gamma}(\mathcal{C})\geq-3$
by equation (3.19). It therefore remains to consider the case where
$e_{\gamma}(\mathcal{C})=1$. In this case $\mathcal{C}_{0}$ does not include
the trivial cylinder $\mathbb{R}\times\gamma$, and $C_{1}$ has exactly one end
asymptotic to a cover of $\gamma$.

If this end is a positive end, then $\gamma$ is not a $p^{-}$-component of
$\alpha$. If $m(\gamma,\alpha)=1$, this is true by definition since $M>1$. If
$m(\gamma,\alpha)>1$, then by Proposition 2.16,
$p^{+}_{\gamma}(m(\gamma,\alpha))=(m(\gamma,\alpha))$, and also $\gamma$ has
irrational rotation number since $\alpha$ is an ECH generator. Then by Lemma
2.18(b), $p_{\gamma}^{-}(m(\gamma,\alpha))\neq(m(\gamma,\alpha))$, so $\gamma$
is not a $p^{-}$-component of $\alpha$. Since $\gamma$ is not a
$p^{-}$-component of $\alpha$, it follows from equation (3.19) that
$S_{\gamma}(\mathcal{C})\geq 0$, so
$T^{\prime}_{\gamma}(\mathcal{C})=S_{\gamma}(\mathcal{C})+2\geq 2$.

Suppose now that the end of $C_{1}$ asymptotic to a cover of $\gamma_{i}$ is
negative. As in the previous paragraph, $\gamma$ is not a $p^{+}$-component of
$\beta$. It then follows from equation (3.19) that
$S_{\gamma}(\mathcal{C})\geq-1$, so $T^{\prime}_{\gamma}(\mathcal{C})\geq 1$.
∎

It follows from equation (3.29) and the above lemma that every $U$-curve
$\mathcal{C}$ satisfies $T^{\prime}(\mathcal{C})\geq-7$.

We can now finish the proof that $|G_{1}|<q^{4/5}$. Note that similarly to
(3.28), we have

(3.30) $\sum_{i=p_{0}+1}^{q}T^{\prime}(\mathcal{C}(i))\leq
4\delta_{2}q^{1/2}/\ell+4\delta_{1}\delta_{2}q^{1/2}.$

Now for $i\in G_{2}$, we have $T(\mathcal{C}(i))=y(\mathcal{C}(i))=0$ by
definition, so $T^{\prime}(\mathcal{C}(i))=0$. Thus

$\begin{split}|G_{1}|&\leq\sum_{i\in G_{1}\sqcup
G_{2}}T^{\prime}(\mathcal{C}(i))\\\
&=\sum^{q}_{i=p_{0}+1}T^{\prime}(\mathcal{C}(i))-\sum_{i\in
G_{0}}T^{\prime}(\mathcal{C}(i))-\sum_{i\in
G_{3}}T^{\prime}(\mathcal{C}(i))\\\ &\leq
4\delta_{2}q^{1/2}/\ell+4\delta_{1}\delta_{2}q^{1/2}+7\left(\delta_{2}q^{1/2}/\epsilon^{\prime}+q^{2/3}\right)\\\
&<q^{4/5}.\end{split}$

Here the second to last line uses (3.25) and Step 2, while the last line uses
the inequality (3.6).

Step 4. We now show that $|G_{2}|<q^{5/6}$.

If $\alpha$ is an orbit set, define $K(\alpha)\leq 0$ to be minus the number
of simple Reeb orbits $\gamma$ such that $m(\gamma,\alpha)>1$. If
$\mathcal{C}\in\mathcal{M}^{J_{k}}(\alpha,\beta)$ is a $U$-curve, define

$K(\mathcal{C})=K(\alpha)-K(\beta)+2y(\mathcal{C}).$

######  Lemma 3.25.

If $i\in G_{2}$, then $K(\mathcal{C}(i))\geq 1$.

###### Proof.

Write $\mathcal{C}=\mathcal{C}(i)\in\mathcal{M}^{J_{k}}(\alpha,\beta)$ where
$\alpha=\alpha(i)$ and $\beta=\alpha(i-1)$. Since $y(\mathcal{C})=0$, we need
to show that $K(\alpha)>K(\beta)$. We will adopt the notation from Lemma 3.22.
By Lemma 3.22(b), the trivial cylinder $\mathbb{R}\times\gamma_{1}$ appears
with multiplicity at least $2$ in $\mathcal{C}_{0}$, while there are no other
trivial cylinders $\mathbb{R}\times\gamma$ in $\mathcal{C}_{0}$ such that
$C_{1}$ has an end asymptotic to a cover of $\gamma$. Thus we can remove any
trivial cylinders other than $\mathbb{R}\times\gamma_{1}$ from
$\mathcal{C}_{0}$ without changing $K(\mathcal{C})$, and we will assume that
this has been done. There are now two cases to consider.

The first case is where $C_{1}$ has two positive ends and one negative end.
Then $m(\gamma_{1},\alpha)>1$, while the other positive end is asymptotic to a
simple Reeb orbit by Lemma 3.22(b)(3), so $K(\alpha)=-1$. On the other hand,
since the trivial cylinder $\mathbb{R}\times\gamma_{1}$ appears with
multiplicity at least $2$ in $\mathcal{C}_{0}$, and the negative end of
$C_{1}$ asymptotic to a cover of $\gamma_{2}\neq\gamma_{1}$ is high
multiplicity, we have $K(\beta)=-2$.

The second case is where $C_{1}$ has one positive end and two negative ends.
Then $\gamma_{1}$ is the only simple Reeb orbit that appears in $\alpha$, so
$K(\alpha)=-1$. On the other hand the distinct simple Reeb orbits
$\gamma_{1},\gamma_{2}$ both appear in $\beta$ with multiplicity greater than
one, so $K(\beta)\in\\{-2,-3\\}$. ∎

We can now complete the proof that $|G_{2}|\leq q^{5/6}$. To start, similarly
to (3.28), we have

(3.31) $\sum_{i=p_{0}+1}^{q}K(\mathcal{C}(i))\leq
2\delta_{2}q^{1/2}/\ell+4\delta_{1}\delta_{2}q^{1/2}.$

Also observe that for every $i$, we have $K(\mathcal{C}(i))\geq-8$ by equation
(2.15). Thus we have

$\begin{split}|G_{2}|&\leq\sum_{i\in G_{2}}K(\mathcal{C}(i))\\\
&=\sum_{i=p_{0}+1}^{q}K(\mathcal{C}(i))-\sum_{i\in G_{0}\sqcup G_{1}\sqcup
G_{3}}K(\mathcal{C}(i))\\\ &\leq
2\delta_{2}q^{1/2}/\ell+4\delta_{1}\delta_{2}q^{1/2}+8\left(\delta_{2}q^{1/2}/\epsilon^{\prime}+q^{4/5}+q^{2/3}\right)\\\
&<q^{5/6}.\end{split}$

Here the second to last line uses Steps 1, 2, and 3, while the last line uses
the inequality (3.7).

Step 5. We now complete the proof of Lemma 3.23. By the decomposition (3.24)
and Steps 1–4, we have

$q-p_{0}<\delta_{2}q^{1/2}/\epsilon^{\prime}+q^{4/5}+q^{5/6}+q^{2/3}.$

This contradicts the inequality (3.8). ∎

####  3.4.2. Conclusion

###### Proof of Proposition 3.1..

Continue with the setup from §3.2.2.

By Lemma 3.23, if $k$ is sufficiently large (we pass to a subsequence so that
this is true for all $k$), there exists a $U$-curve for $(\lambda_{k},J_{k})$
such that the positive orbit set has action $\leq T_{q}$, and the nontrivial
component is a low energy cylinder. Choose such a $U$-curve for each $k$, and
denote the nontrivial component by $C_{k}$. Let $\alpha_{k}$ and $\beta_{k}$
denote the Reeb orbits for $\lambda_{k}$ to which the positive and negative
ends of $C_{k}$ respectively are asymptotic.

Similarly to §3.2.2, it follows from Lemma 3.5 that for each $k$, there are
unique Reeb orbits $\underline{\alpha_{k}}$ and $\underline{\beta_{k}}$ for
$\lambda$ that are homotopic to $\alpha_{k}$ and $\beta_{k}$ respectively in
$N$. Since there is a uniform upper bound on the action of $\alpha_{k}$ and
$\beta_{k}$, there are only finitely many options for $\alpha_{k}$ and
$\beta_{k}$. We can then pass to a subsequence so that there are Reeb orbits
$\alpha_{\infty}$ and $\beta_{\infty}$ for $\lambda$ such that
$\underline{\alpha_{k}}=\alpha_{\infty}$ and
$\underline{\beta_{k}}=\beta_{\infty}$ for each $k$. Then the sequence
$((\lambda_{k},J_{k},\alpha_{k},\beta_{k},C_{k}))$ satisfies all of the bullet
points in Proposition 3.1, as well as assertions (a) and (b). To complete the
proof of Proposition 3.1, we now prove assertions (c) and (d).

To prove assertion (c) in Proposition 3.1, note that by Lemma 3.10, if $k$ is
sufficiently large (we pass to a subsequence so that this is true for all
$k$), then

(3.32) $\mathcal{A}(\alpha_{k})-\mathcal{A}(\beta_{k})>\overline{h}.$

It then follows from Lemma 3.11 that the Reeb orbits $\alpha_{k}$ and
$\beta_{k}$ have multiplicity $\geq M\geq 3$. In particular, since $C_{k}$
together with some trivial cylinders goes between ECH generators, the orbits
$\alpha_{k}$ and $\beta_{k}$ are elliptic and so their rotation numbers are
irrational.

Let $m$ denote the covering multiplicity of the Reeb orbit $\alpha_{k}$, let
$\gamma$ denote the underlying simple Reeb orbit, and let
$\theta\in\mathbb{R}/\mathbb{Z}$ denote the rotation number of $\gamma$, so
that the rotation number of $\alpha_{k}$ is $m\theta$. By Proposition 2.16, we
have $p_{\gamma}^{+}(m)=(m)$. Now suppose that $\alpha_{\infty}$ is
degenerate; then its rotation number must be $0\in\mathbb{R}/\mathbb{Z}$. Thus
the rotation number of the simple orbit underlying $\alpha_{\infty}$ is a
rational number, which we write in lowest terms as $a/b$, and $m$ is an
integer multiple of $b$. Choose $\varepsilon>0$ sufficiently small that
Example 2.15 is applicable, and also sufficiently small so that
$m\varepsilon<1/2$. By Lemma 3.5, if $k$ is sufficiently large (we pass to a
subsequence so that this is true for all $k$), then
$|\theta-a/b|<\varepsilon$. By Example 2.15, if
$\theta-a/b\in(0,\varepsilon)$, then $p_{\gamma}^{+}(m)=(m/b,\ldots,m/b)$.
Since $m\geq M$ and we assumed in §3.2.5 that $M>B_{0}$, we have $m>b$, so the
partition $p_{\gamma}^{+}(m)$ has at least two elements. This contradicts the
fact that $p_{\gamma}^{+}(m)=(m)$. Thus we must have
$\theta-a/b\in(-\varepsilon,0)$, and it follows that the rotation number of
$\alpha_{k}$ is $m\theta\in(-1/2,0)$.

A symmetric argument shows that if $\beta_{\infty}$ is degenerate, then if $k$
is sufficiently large (we pass to a subsequence so that this is true for all
$k$), then the rotation number of $\beta_{k}$ is in the interval $(0,1/2)$.

Finally, to prove assertion (d) in Proposition 3.1, we will invoke Proposition
2.22. We already know that $C_{1}$ satisfies all of the criteria in
Proposition 2.22 except possibly for (e), which we still need to show. That
is, we need to show that the component of the moduli space
$\mathcal{M}^{J_{k}}(\alpha_{k},\beta_{k})/\mathbb{R}$ containing $C_{1}$ is
compact. To do so, we use a variant of the proof of [9, Lem. 4.10(c)].

Suppose to get a contradiction that this moduli space is not compact. Then
there must exist a sequence of elements of the moduli space that converges to
a $J_{k}$-holomorphic building from $\alpha_{k}$ to $\beta_{k}$ with at least
two levels. No level of the building can include a holomorphic plane, because
the minimum action of a Reeb orbit is greater than $\epsilon^{\prime}$ which
is greater than $\mathcal{A}(\alpha)-\mathcal{A}(\beta)$. Hence every level of
the building is a cylinder. By Proposition 2.6, there are exactly two levels,
and each level has ECH index $1$. Let $\gamma$ denote the intermediate Reeb
orbit. Since $\alpha_{k}$ and $\beta_{k}$ are elliptic, it follows from the
ECH index parity conditions discussed in §2.3 that $\gamma$ is a cover of a
positive hyperbolic orbit. By the partition conditions in Proposition 2.16 and
Example 2.14, the Reeb orbit $\gamma$ must be simple. By the inequality
(3.32), at least one of the two cylinders in the building must have the
property that the action difference between the Reeb orbits at the positive
and negative ends is greater than $\overline{h}/2$. It then follows from Lemma
3.11 that $\mathcal{A}(\alpha)-\mathcal{A}(\beta)\geq\epsilon^{\prime}$, which
contradicts the fact that the cylinder $C_{k}$ is low energy. ∎

##  4\. Preliminaries about holomorphic curves in symplectizations

Equipped with our Birkhoff sections for the nondegenerate perturbations
$\lambda_{n}$, we now turn to the rest of the proof. Before proceeding, we
first explain the general idea behind our argument.

#### Outline of the argument

We have a sequence of holomorphic cylinders $C_{n}$ that give a Birkhoff
section for $\lambda_{n}$ and we would like to prove a two or infinity result
in the limit. Our basic idea for this is to obtain a compact moduli space of
holomorphic cylinders in the limit. To prove this, we first prove a general
automatic transversality result that holds even for degenerate contact forms
as long as the curves under consideration are asymptotic to Reeb orbits at
their punctures and converge exponentially fast to them; we leverage a
weighted Fredholm theory for this. Next, we need to show that the $C_{n}$ give
rise to curves with the required exponential convergence; this is where
condition (c) in Proposition 3.1 is used. We also obtain a kind of weak
version of the usual SFT compactness theorem that suffices for our purposes.
Thus, we are able to obtain a well-defined nonempty transverse moduli space of
cylinders in the limit. To show that it is compact, the basic idea is as
follows: a nonconvergent sequence of cylinders would give rise to an orbit as
an asymptotic limit with positive intersection number with the relative
homology class of the projection of the $C_{n}$; thus, close to the limit we
would obtain a loop in the projection of the image of $C_{n}$ with positive
intersection number, and this contradicts the fact that each $C_{n}$ has an
embedded projection to the 3-manifold. Equipped with the necessary moduli
space of cylinders in the limit, we then lift the Reeb flow to the moduli
space via an evaluation map; we show that the lifted flow has an annular
global cross-section, and so we are justified in applying a theorem of Franks
to conclude that if the original Reeb flow had only finitely many orbits, this
lifted flow would have none, hence the original flow has precisely two.

Executing the above sketch will take up the remainder of our paper. To
proceed, let $\lambda$ be a contact form on a closed $3$-manifold $Y$, with
Reeb vector field $R$ and contact structure $\xi=\ker\lambda$. We do not make
nondegeneracy assumptions on $\lambda$. Denote by

$\pi_{\lambda}:TY\to\xi$

the projection along $R$.

###  4.1. Generalities

We begin by setting notation and collecting some known results. In what
follows, it will be convenient to write Reeb orbits as equivalence classes of
pairs $(x,T)$, where $x:\mathbb{R}\to Y$ is a periodic trajectory of $R$, and
$T>0$ is a period of $x$. It will also be convenient to make a choice of
marked point on the images of all simple Reeb orbits; this uniquely determines
$x$ by requiring that $x(0)$ is the marked point. We will also want to define
two different Reeb orbits as being geometrically distinct if their images do
not intersect.

Given a Reeb orbit $\gamma=(x,T)$, the bundle
$x(T\cdot)^{*}\xi\to\mathbb{R}/\mathbb{Z}$ will be denoted by $\xi_{\gamma}$
from now on. Let $J:\xi\to\xi$ be a $d\lambda$-compatible complex structure;
from this point onwards, we will want to distinguish between the complex
structure on $\xi$ and the corresponding $\lambda$-compatible almost complex
structure on the symplectization, so we will denote the latter by $\tilde{J}$.
The associated asymptotic operator of $\gamma=(x,T)$, acting on sections of
$\xi_{\gamma}$, is defined by

$A_{\gamma}:\eta\mapsto-J(\nabla_{t}\eta-\nabla_{\eta}R)$

where $\nabla$ is any choice of symmetric connection on $Y$, and $\nabla_{t}$
denotes the induced covariant derivative of vector fields along the curve
$t\mapsto x(Tt)$. Note that $A_{\gamma}$ is independent of choice of
connection, and that $\gamma$ is nondegenerate precisely when $0$ is not an
eigenvalue of $A_{\gamma}$. Note also that $A_{\gamma}$ depends on $J$ but we
do not make this dependence explicit in the notation, for the sake of
simplicity. The operator $A_{\gamma}$ defines a self-adjoint unbounded
operator that acts on $W^{1,2}$-sections of $\xi_{\gamma}$ whose spectrum
$\sigma(A_{\gamma})\subset\mathbb{R}$ is discrete, consists only of
eigenvalues, and accumulates only at $\pm\infty$. If
$\nu\in\sigma(A_{\gamma})$ then every eigenvector for $\nu$ is nowhere
vanishing and has the same winding number in a given symplectic trivialization
$\tau$ of $\xi_{\gamma}$. We denote this winding number by

${\mathrm{wind}}_{\tau}(\nu)\in\mathbb{Z}\,.$

It is useful to note the following. First of all, there are two eigenvalues
with a given winding number, counted with multiplicities. Next,

$\nu_{1}\leq\nu_{2}\Rightarrow{\mathrm{wind}}_{\tau}(\nu_{1})\leq{\mathrm{wind}}_{\tau}(\nu_{2}),$

and, moreover, any integer is the winding number of some eigenvalue.

Given $\delta\in\mathbb{R}$, we denote by $\nu^{<\delta}(\gamma)$ the largest
eigenvalue $\nu$ satisfying $\nu<\delta$, and by $\nu^{\geq\delta}(\gamma)$
the smallest eigenvalue $\nu$ satisfying $\nu\geq\delta$. Set

$\displaystyle
p^{\delta}(\gamma)={\mathrm{wind}}_{\tau}(\nu^{\geq\delta}(\gamma))-{\mathrm{wind}}_{\tau}(\nu^{<\delta}(\gamma))\in\\{0,1\\}\,.$

The $\delta$-weighted Conley-Zehnder index of $\gamma$ in the trivialization
$\tau$ is

(4.1) ${\rm
CZ}^{\delta}_{\tau}(\gamma)=2{\mathrm{wind}}_{\tau}(\nu^{<\delta}(\gamma))+p^{\delta}(\gamma)\,.$

When $\delta=0$ we simply write ${\rm CZ}_{\tau}(\gamma)$. See [21, Section 3]
for more details. When $\delta=0$ and $\gamma$ is nondegenerate, this agrees
with (2.1).

######  Remark 4.1 (Martinet tubes).

Consider a periodic orbit $\gamma=(x,T)$ with primitive period $T_{0}$. A
Martinet tube around $\gamma$ is a smooth diffeomorphism
$\Psi:U\to\mathbb{R}/\mathbb{Z}\times B$, where $U$ is an open neighborhood of
$x(\mathbb{R})$ and $B\subset\mathbb{C}$ is an open ball centered at the
origin, such that $\Psi(x(T_{0}t))=(t,0)$ and
$\Psi_{*}\lambda=f(\theta,x_{1}+ix_{2})(d\theta+x_{1}dx_{2})$, for some
function $f$ satisfying $f|_{\mathbb{R}/\mathbb{Z}\times\\{0\\}}\equiv T_{0}$
and $df|_{\mathbb{R}/\mathbb{Z}\times\\{0\\}}\equiv 0$.

From now on we fix a closed connected orientable surface $S$, and

$\Gamma_{+}=\\{z_{1}^{+},\dots,z_{m_{+}}^{+}\\}\qquad\Gamma_{-}=\\{z_{1}^{-},\dots,z_{m_{-}}^{-}\\}$

disjoint ordered subsets of $S$. Denote $\Gamma=\Gamma_{+}\cup\Gamma_{-}$,
$\dot{S}=S\setminus\Gamma$. Once a conformal structure $j$ on $S$ is chosen,
for any $z_{*}\in\Gamma$ and any choice of holomorphic embedding
$\psi:(\mathbb{D},i,0)\to(S,j,z_{*})$ we call the map

$(s,t)\in[0,+\infty)\times\mathbb{R}/\mathbb{Z}\mapsto\psi(e^{-2\pi(s+it)})\in\psi(\mathbb{D}\setminus\\{0\\})$

positive holomorphic polar coordinates centered at $z_{*}$. Similarly, we call

$(s,t)\in(-\infty,0]\times\mathbb{R}/\mathbb{Z}\mapsto\psi(e^{2\pi(s+it)})\in\psi(\mathbb{D}\setminus\\{0\\})$

negative holomorphic polar coordinates centered at $z_{*}$.

We sometimes call a ${\widetilde{J}}$-holomorphic map a pseudo-holomorphic map
when there is no need to refer to ${\widetilde{J}}$. Consider a conformal
structure $j$ on $S$, and a nonconstant pseudo-holomorphic map

(4.2) ${\widetilde{u}}=(a,u):(\dot{S},j)\to(\mathbb{R}\times
Y,{\widetilde{J}})$

with finite Hofer energy

$0<E({\widetilde{u}})=\sup_{\phi}\int_{\dot{S}}{\widetilde{u}}^{*}d(\phi\lambda)<\infty\,.$

The supremum is taken over all smooth $\phi:\mathbb{R}\to[0,1]$ satisfying
$\phi^{\prime}\geq 0$. It is proved in [19] that for every $z_{*}\in\Gamma$
exactly one of the following holds:

  * •

$z_{*}$ is a positive puncture: $a(z)\to+\infty$ as $z\to z_{*}$.

  * •

$z_{*}$ is a negative puncture: $a(z)\to-\infty$ as $z\to z_{*}$.

  * •

$z_{*}$ is a removable puncture: ${\widetilde{u}}$ can be smoothly continued
across $z_{*}$.

We proceed assuming that $\Gamma_{+}$ consists of positive punctures and
$\Gamma_{-}$ consists of negative punctures. In what follows we will refer to
the sign $\epsilon=\pm 1$ of a puncture according to the above classification,
i.e. punctures in $\Gamma_{+}$ have sign $\epsilon=+1$ and punctures in
$\Gamma_{-}$ have sign $\epsilon=-1$. We also consider positive holomorphic
polar coordinates centered at each positive puncture, and negative holomorphic
polar coordinates centered at each negative puncture. These sets of
coordinates will be all denoted by $(s,t)$, with no fear of ambiguity. All
definitions and statements below are independent of these choices. We write
${\widetilde{u}}(s,t)$ to denote the map ${\widetilde{u}}$ near a puncture
written in holomorphic polar coordinates.

######  Theorem 4.2 (Hofer [19]).

Consider a puncture of sign $\epsilon$, and any sequence $s_{n}\to+\infty$.
There exist a subsequence $s_{n_{k}}$, a Reeb orbit $\gamma=(x,T)$ and
$t_{0}\in\mathbb{R}/\mathbb{Z}$ such that $u(\epsilon s_{n_{k}},t+t_{0})\to
x(Tt)$ in $C^{\infty}(\mathbb{R}/\mathbb{Z},Y)$ as $k\to\infty$.

######  Definition 4.3.

Consider a puncture $z_{*}$ of sign $\epsilon$.

  * •

We say that $\gamma=(x,T)$ is an asymptotic limit of ${\widetilde{u}}$ at
$z_{*}$ if we can find $s_{n}\to+\infty$ and $t_{0}\in\mathbb{R}/\mathbb{Z}$
such that $u(\epsilon s_{n},t+t_{0})\to x(Tt)$ in
$C^{0}(\mathbb{R}/\mathbb{Z},Y)$ as $n\to\infty$.

  * •

We say that ${\widetilde{u}}$ is asymptotic to $\gamma=(x,T)$ at $z_{*}$ if we
can find $t_{0}\in\mathbb{R}/\mathbb{Z}$ such that $u(s,t+t_{0})\to x(Tt)$ in
$C^{0}(\mathbb{R}/\mathbb{Z},Y)$ as $\epsilon s\to+\infty$.

######  Remark 4.4.

Asymptotic limits need not be unique, see [46]. To say that every asymptotic
limit of ${\widetilde{u}}$ at $z_{*}$ is equal to $\gamma$ is not the same as
saying that ${\widetilde{u}}$ is asymptotic to $\gamma$ at $z_{*}$.

For the next definition, assume that ${\widetilde{u}}$ is asymptotic to
$\gamma=(x,T)$ at a puncture $z_{*}$. Let $m\in\mathbb{N}$ be the covering
multiplicity of $\gamma$, and $\epsilon$ be the sign of the puncture at
$z_{*}$. Choose $\Psi:U\to\mathbb{R}/\mathbb{Z}\times B$ a Martinet tube
around $\gamma$. If $\epsilon s$ is large enough then $u(s,t)\in U$ and we can
write in components

$\Psi\circ u(s,t)=(\theta(s,t),z(s,t))\,.$

######  Definition 4.5.

We say that ${\widetilde{u}}$ has an asymptotic formula at $z_{*}$ if either
$z(s,t)$ vanishes identically, or there exist $b>0$,
$\mu\in\sigma(A_{\gamma})$ such that $\epsilon\mu<0$ and the following holds:

  * •

For some $a_{0},t_{0}\in\mathbb{R}$ and some lift
$\tilde{\theta}:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ of $\theta(s,t)$

$\lim_{\epsilon s\to+\infty}\sup_{t\in\mathbb{R}/\mathbb{Z}}\ e^{b\epsilon
s}\left(|D^{\beta}[a(s,t)-Ts-
a_{0}]|+|D^{\beta}[\tilde{\theta}(s,t+t_{0})-mt]|\right)=0$

for every $\beta=(\beta_{1},\beta_{2})\in\mathbb{N}_{0}\times\mathbb{N}_{0}$.

  * •

There is an eigenvector of $\mu$, represented as a function
$v:\mathbb{R}/\mathbb{Z}\to\mathbb{C}$ in the frame
$\\{\partial_{x_{1}},\partial_{x_{2}}\\}$ of $\xi$ along $\gamma$, such that

(4.3) $z(s,t+t_{0})=e^{\mu s}(v(t)+\Delta(s,t))$

for some $\Delta(s,t)$ satisfying $\sup_{t}|D^{\beta}\Delta(s,t)|\to 0$ as
$\epsilon s\to+\infty$, for every multi-index
$\beta=(\beta_{1},\beta_{2})\in\mathbb{N}_{0}\times\mathbb{N}_{0}$.

The eigenvalue $\mu$ is called the asymptotic eigenvalue of ${\widetilde{u}}$
at $z_{*}$, and the eigenvector in (4.3) is called the asymptotic eigenvector
of ${\widetilde{u}}$ at $z_{*}$. We say that ${\widetilde{u}}$ has a
nontrivial asymptotic formula at $z_{*}$ if it has an asymptotic formula at
$z_{*}$ and $z(s,t)$ does not vanish identically.

The above definition does not depend on the choices of holomorphic polar
coordinates and Martinet tube.

######  Theorem 4.6 (Hofer, Wysocki and Zehnder [20]).

If an asymptotic limit $\gamma$ of ${\widetilde{u}}$ at the puncture $z_{*}$
is a nondegenerate periodic orbit then ${\widetilde{u}}$ is asymptotic to
$\gamma$ at $z_{*}$, and has an asymptotic formula at $z_{*}$.

Choose any Riemannian metric on $Y$, and denote the associated exponential map
by $\exp$.

######  Definition 4.7 ([25]).

Let $z_{*}$ be a puncture, and let $\epsilon$ be its sign. We say that $z_{*}$
is a nondegenerate puncture if ${\widetilde{u}}$ is asymptotic to a Reeb orbit
$\gamma=(x,T)$ at $z_{*}$, i.e. $u(s,\cdot)\to x(T(\cdot+t_{0}))$ in $C^{0}$
as $\epsilon s\to+\infty$ for some $t_{0}$, and:

  * •

There exists $a_{0}\in\mathbb{R}$ such that $\sup_{t}|a(s,t)-Ts-a_{0}|$ as
$\epsilon s\to+\infty$.

  * •

If $\pi_{\lambda}\circ du$ does not vanish identically, then
$\pi_{\lambda}\circ du(s,t)\neq 0$ for $\epsilon s\gg 1$.

  * •

Let $\zeta(s,t)\in T_{x(T(t+t_{0}))}Y$ be defined by
$\exp(\zeta(s,t))=u(s,t)$, $\epsilon s\gg 1$. There exists $b>0$ such that
$\sup_{t}e^{b\epsilon s}|\zeta(s,t)|\to 0$ as $\epsilon s\to+\infty$.

We emphasize that the Reeb orbit associated to a nondegenerate puncture need
not be nondegenerate. Obviously, if ${\widetilde{u}}$ has an asymptotic
formula at $z_{*}$ then $z_{*}$ is a nondegenerate puncture of
${\widetilde{u}}$. The converse is the content of the following statement.

######  Theorem 4.8 ([25], Corollary 6.6).

If $z_{*}$ is a nondegenerate puncture of ${\widetilde{u}}$, then
${\widetilde{u}}$ has an asymptotic formula at $z_{*}$.

Theeorem 4.8 is a consequence of arguments originally given by Hofer, Wysocki
and Zehnder [20] to prove Theorem 4.6. The form of the asymptotic formula
(4.3) presented above first appeared in [45]. It should be remarked that the
above discussion has a version in higher dimensions.

###  4.2. Weighted Fredholm theory and automatic transversality

In this section, we prove what we will need to know about the Fredholm theory
for a pseudo-holomorphic curve with exponential convergence to Reeb orbits at
its punctures; we do not assume that the Reeb orbits at these punctures are
nondegenerate.

We assume in this section that ${\widetilde{u}}$ is a pseudo-holomorphic map
as in (4.2) with an asymptotic formula at every puncture. Assume also that
$\pi_{\lambda}\circ du$ does not vanish identically on $\dot{S}$. By
connectedness of $S$, the latter is equivalent to saying that
${\widetilde{u}}$ is not a (possibly branched) cover of a trivial cylinder. It
follows from Carleman’s similarity principle, and the fact that
$\pi_{\lambda}\circ du$ satisfies a Cauchy-Riemann type equation in suitable
local representations, that every zero of $\pi_{\lambda}\circ du$ is isolated
and contributes positively to the total algebraic count. By Theorem 4.8 and
Definition 4.5, it follows that ${\widetilde{u}}$ has a nontrivial asymptotic
formula at every puncture, and that there are only finitely many zeros of
$\pi_{\lambda}\circ du$. Denote by $\gamma^{\pm}_{i}$ the asymptotic limit of
${\widetilde{u}}$ at $z^{\pm}_{i}$, and by
$\mu^{\pm}_{i}\in\sigma(A_{\gamma^{\pm}_{i}})$ the asymptotic eigenvalue of
${\widetilde{u}}$ at $z^{\pm}_{i}$. Note that $\mu^{+}_{i}<0$ and
$\mu^{-}_{i}>0$.

The invariants ${\mathrm{wind}}_{\pi}$ and ${\mathrm{wind}}_{\infty}$ were
introduced in [21]. The invariant ${\mathrm{wind}}_{\pi}({\widetilde{u}})$ is
defined as the total algebraic count of zeros of $\pi_{\lambda}\circ du$. In
particular

(4.4)
${\mathrm{wind}}_{\pi}({\widetilde{u}})=0\qquad\Rightarrow\qquad\text{$u$ is
an immersion transverse to $R$.}$

A global symplectic trivialization $\tau$ of $u^{*}\xi$ induces symplectic
trivializations $\tau^{\pm}_{i}$ on $\xi_{\gamma^{\pm}_{i}}$, up to homotopy.
The invariant

(4.5)
${\mathrm{wind}}_{\infty}({\widetilde{u}})=\sum_{i=1}^{m_{+}}{\mathrm{wind}}_{\tau^{+}_{i}}(\mu^{+}_{i})-\sum_{i=1}^{m_{-}}{\mathrm{wind}}_{\tau^{-}_{i}}(\mu^{-}_{i})$

is independent of the choice of $\tau$. Standard degree theory, see [21],
shows that

(4.6)
${\mathrm{wind}}_{\pi}({\widetilde{u}})={\mathrm{wind}}_{\infty}({\widetilde{u}})-2+2g+\\#\Gamma$

where $g$ is the genus of $S$.

Choose $\delta>0$ small enough such that

(4.7) $[-\delta,0)\cap\sigma(A_{\gamma^{+}_{i}})=\emptyset\ \forall
i\in\\{1,\dots,m^{+}\\},\qquad(0,\delta]\cap\sigma(A_{\gamma^{-}_{i}})=\emptyset\
\forall i\in\\{1,\dots,m^{-}\\}\,.$

Hence $\mu^{+}_{i}<-\delta$ and $\mu^{-}_{i}>\delta$. We consider a
$\delta$-weighted Fredholm index defined as

(4.8)
${\mathrm{ind}}_{\delta}({\widetilde{u}})=-2+2g+m_{+}+m_{-}+\sum_{i=1}^{m_{+}}{\rm
CZ}_{\tau^{+}_{i}}^{-\delta}(\gamma^{+}_{i})-\sum_{i=1}^{m_{-}}{\rm
CZ}_{\tau^{-}_{i}}^{\delta}(\gamma^{-}_{i})\,.$

Denote $\Gamma_{+}^{\rm odd}=\\{z^{+}_{i}\mid{\rm
CZ}^{-\delta}_{\tau^{+}_{i}}(\gamma^{+}_{i})\ \text{is odd}\\}$, and
$\Gamma_{-}^{\rm odd}=\\{z^{-}_{i}\mid{\rm
CZ}^{\delta}_{\tau^{-}_{i}}(\gamma^{-}_{i})\ \text{is odd}\\}$. These sets are
independent of the choice of $\delta$ with the properties above, and also
independent of $\tau$. A puncture in $\Gamma^{\rm odd}=\Gamma_{+}^{\rm
odd}\cup\Gamma_{-}^{\rm odd}$ will be called an odd puncture.

######  Lemma 4.9.