by contact homology, in the sense that their definition and the proof of their
properties require only compactness results for pseudoholomorhpic curves.
Their computation involves some knowledge of Gromov–Witten invariants, see
Proposition 10.3 below.

Before discussing them, we recall some basic facts and fix notation concerning
the symplectization of a contact manifold. Let $\xi$ be a co-oriented contact
structure on the $(2n-1)$-dimensional manifold $M$. A cotangent vector $x\in
T^{*}_{p}M$ is said to define $\xi(p)$ if $\ker x=\xi(p)$ and $x$ is positive
on vectors in $T_{p}M$ which are positively transverse to $\xi(p)$. We denote
by

$\widetilde{M}\coloneqq\\{x\in T^{*}M\mid x\mbox{ defines }\xi(\pi(x))\\}$

the symplectization of $M$, where $\pi:T^{*}M\rightarrow M$ denotes the
footpoint projection. The symplectization $\widetilde{M}$ is a
$2n$-dimensional submanifold of $T^{*}M$. It carries a Liouville $1$-form
$\widetilde{\lambda}$ and a symplectic form
$\widetilde{\omega}=d\widetilde{\lambda}$ given by the restriction of the
standard Liouville $1$-form and symplectic form of $T^{*}M$. Moreover,
$\widetilde{M}$ carries a canonical $\mathbb{R}$-action given by

$\mathbb{R}\times\widetilde{M}\rightarrow\widetilde{M},\qquad(a,\eta)\mapsto
e^{a}\eta.$

We denote by $\partial_{a}$ the vector field on $\widetilde{M}$ generating
this action. The choice of a contact form $\alpha\in\mathcal{F}(M,\xi)$
induces the diffeomorphism

$\varphi_{\alpha}:\mathbb{R}\times
M\rightarrow\widetilde{M},\qquad(a,p)\mapsto e^{a}\alpha(p),$

such that $\varphi_{\alpha}^{*}\widetilde{\omega}=d(e^{a}\alpha)$ where $a$ is
the variable in the first factor of $\mathbb{R}\times M$. Under the
identification of $\widetilde{M}$ with $\mathbb{R}\times M$ defined by
$\varphi_{\alpha}$, the canonical action of $\mathbb{R}$ is given by
translations in the first factor (this justifies the choice of denoting the
generating vector field by $\partial_{a}$) and the contact-type hypersurface

$M_{\alpha}\coloneqq\\{\alpha(p)\mid p\in M\\}\subset\widetilde{M}$

corresponds to the set $\\{a=0\\}$. The contact form $\alpha$ and the Reeb
vector field $R_{\alpha}$ can be seen as a translation invariant 1-form and
vector field on $\mathbb{R}\times M$, and by the diffeomorphism
$\varphi_{\alpha}$ induce an $\mathbb{R}$-invariant 1-form and an
$\mathbb{R}$-invariant vector field on $\widetilde{M}$:

$\widetilde{\alpha}:={\varphi_{\alpha}}_{*}\alpha,\qquad\widetilde{R}_{\alpha}:={\varphi_{\alpha}}_{*}R_{\alpha}.$

The vector field $\widetilde{R}_{\alpha}$ is tangent to the hypersurface
$M_{\alpha}$ and all its images by the canonical $\mathbb{R}$-action, and
spans the characteristic distribution of these hypersurfaces.

Given contact forms $\alpha,\beta\in\mathcal{F}(M,\xi)$ with $\alpha<\beta$,
we denote by $\widetilde{M}_{\alpha}^{\beta}\subset\widetilde{M}$ the
symplectic cobordism which is given by the region in $\widetilde{M}$ bounded
by $M_{\alpha}$ and $M_{\beta}$, that is,

$\widetilde{M}_{\alpha}^{\beta}\coloneqq\\{x\in\widetilde{M}\mid\alpha(\pi(x))<x<\beta(\pi(x))\mbox{\scalebox{0.9}{
on vectors which are positively transverse to }}\xi(\pi(x))\\}.$

The diffeomorphism $\varphi_{\beta}$ exhibits the region
$\widetilde{M}_{\beta}^{+}\subset\widetilde{M}$ above the hypersurface
$M_{\beta}$ as a positive cylindrical end attached to the symplectic cobordism
$\widetilde{M}_{\alpha}^{\beta}$. Similarly, we can regard the region
$\widetilde{M}_{\alpha}^{-}\subset\widetilde{M}$ below the hypersurface
$M_{\alpha}$ as a negative cylindrical end of $\widetilde{M}_{\alpha}^{\beta}$
via the diffeomorphism $\varphi_{\alpha}$.

Let us recall the following terminology from [BEH+03]. An almost complex
structure $J$ on $\widetilde{M}$ is called adjusted to a contact form
$\alpha\in\mathcal{F}(M,\xi)$ if it satisfies the following properties:

  1. 1.

$J$ is invariant under the canonical $\mathbb{R}$-action on $\widetilde{M}$.

  2. 2.

$J\partial_{a}=\widetilde{R}_{\alpha}$.

  3. 3.

The codimension-two distribution
$\xi_{\alpha}:={\varphi_{\alpha}}_{*}((0)\times\xi)$ is invariant under $J$.
Moreover, the restriction of $J$ to $\xi_{\alpha}$ is compatible with
$d\widetilde{\alpha}$.

Suppose that $J$ is an $\alpha$-adjusted almost complex structure on
$\widetilde{M}$. Let $u:(\Sigma,j)\rightarrow(\widetilde{M},J)$ be a
$J$-holomorphic map whose domain is a punctured Riemann surface. We define the
action $\mathcal{A}_{\alpha}(u)$ to be

$\mathcal{A}_{\alpha}(u)\coloneqq\int_{\Sigma}u^{*}d\widetilde{\alpha}\in[0,+\infty].$

Note that since $u$ is $J$-holomorphic and $J$ is adjusted to $\alpha$, the
$2$-form $u^{*}d\widetilde{\alpha}$ is everywhere non-negative on $\Sigma$.
Thus $\mathcal{A}_{\alpha}(u)$ is indeed well-defined an non-negative.
Moreover, we define

$E_{\alpha}(u)\coloneqq\sup_{\rho}\int_{\Sigma}u^{*}\bigl{(}\rho\circ
a_{\alpha}\,da_{\alpha}\wedge\widetilde{\alpha}\bigr{)}\in[0,+\infty],\qquad\mbox{with
}a_{\alpha}:=a\circ\varphi_{\alpha}^{-1},$

where the supremum is taken over all compactly supported functions
$\rho:\mathbb{R}\rightarrow\mathbb{R}_{\geq 0}$ whose integral is equal to
$1$. As before, the $2$-form $u^{*}(\rho\circ
a_{\alpha})da_{\alpha}\wedge\widetilde{\alpha}$ is everywhere non-negative
because $J$ is $\alpha$-adjusted and $u$ is $J$-holomorphic. We say that $u$
is a finite-energy curve if both $\mathcal{A}_{\alpha}(u)$ and $E_{\alpha}(u)$
are finite. If $u$ is a finite-energy curve and $\alpha$ is non-degenerate,
then $u$ is positively or negatively asymptotic to trivial cylinders over
periodic Reeb orbits of $\alpha$ at every puncture of $\Sigma$ which is not a
removable singularity of $u$. We define the positive action
$\mathcal{A}_{\alpha}^{+}(u)$ to be the sum of the actions (with respect to
$\alpha$) of the positive asymptotic Reeb orbits of $u$. Similarly, we define
the negative action $\mathcal{A}_{\alpha}^{-}(u)$ to be the sum of the actions
of the negative asymptotic Reeb orbits of $u$. By Stokes’ theorem, we have
$\mathcal{A}_{\alpha}(u)=\mathcal{A}_{\alpha}^{+}(u)-\mathcal{A}_{\alpha}^{-}(u)$.

Let $\alpha<\beta$ be contact forms and assume that $J$ is a compatible almost
complex structure on $\widetilde{M}$ whose restriction to
$\widetilde{M}_{\beta}^{+}$ is $\beta$-adjusted and whose restriction to
$\widetilde{M}_{\alpha}^{-}$ is $\alpha$-adjusted. We define the action

$\mathcal{A}_{\alpha}^{\beta}(u)\coloneqq\int_{u^{-1}(\widetilde{M}_{\beta}^{+})}u^{*}d\widetilde{\beta}+\int_{u^{-1}(\widetilde{M}_{\alpha}^{\beta})}u^{*}\widetilde{\omega}+\int_{u^{-1}(\widetilde{M}_{\alpha}^{-})}u^{*}d\widetilde{\alpha}\in[0,+\infty].$

All three summands in this expression are well-defined and non-negative. For
the terms involving the cylindrical ends $\widetilde{M}_{\beta}^{+}$ and
$\widetilde{M}_{\alpha}^{-}$ this follows from the fact that $J$ is adjusted
and for the cobordism $\widetilde{M}_{\alpha}^{\beta}$ this uses that $J$ is
compatible. In addition, we define

$E_{\alpha}^{\beta}(u)\coloneqq\sup_{\rho^{+}}\int_{u^{-1}(\widetilde{M}_{\beta}^{+})}u^{*}\bigl{(}\rho^{+}\circ
a_{\beta}\,da_{\beta}\wedge\widetilde{\beta}\bigr{)}+\sup_{\rho^{-}}\int_{u^{-1}(\widetilde{M}_{\alpha}^{-})}u^{*}\bigl{(}\rho^{-}\circ
a_{\alpha}\,da_{\alpha}\wedge\widetilde{\alpha}\bigr{)}\in[0,+\infty].$

Here the suprema are taken over all compactly supported, non-negative
functions $\rho^{\pm}$ on $\mathbb{R}_{\geq 0}$ and $\mathbb{R}_{\leq 0}$,
respectively, with integral equal to $1$. Again, a curve $u$ is said to have
finite energy if both $\mathcal{A}_{\alpha}^{\beta}(u)$ and
$E_{\alpha}^{\beta}(u)$ are finite. If $u$ has finite energy and $\alpha$ and
$\beta$ are non-degenerate, then $u$ is positively asymptotic to periodic Reeb
orbits of $\beta$ or negatively asymptotic to periodic Reeb orbits of $\alpha$
at punctures which are not removable singularites. The positive action
$\mathcal{A}_{\beta}^{+}(u)$ and the negative action
$\mathcal{A}_{\alpha}^{-}(u)$ are defined as before and we have
$\mathcal{A}_{\alpha}^{\beta}(u)=\mathcal{A}_{\beta}^{+}(u)-\mathcal{A}_{\alpha}^{-}(u)$
by Stokes’ theorem.

Given a contact form $\alpha\in\mathcal{F}(M,\xi)$, define
$\mathcal{J}(\alpha)$ to be the set all compatible almost complex structures
$J$ on the symplectization $\widetilde{M}$ with the property that there exist
real numbers $s^{-}<s^{+}$ such that $J$ is adjusted to $\alpha$ on both
$\widetilde{M}_{e^{s^{\pm}}\alpha}^{\pm}$. Here the numbers $s^{\pm}$ are
allowed to depend on $J$. For a given $J\in\mathcal{J}(\alpha)$, there are
many choices of $s^{\pm}$ making it adjusted to $\alpha$ on
$\widetilde{M}_{e^{s^{\pm}}\alpha}^{\pm}$. While the action
$\mathcal{A}^{e^{s^{+}}\alpha}_{e^{s^{-}}\alpha}$ and the energy
$E^{e^{s^{+}}\alpha}_{e^{s^{-}}\alpha}$ themselves depend on the choice of
$s^{\pm}$, the property of being a finite-energy $J$-holomorphic curve does
not.

Suppose now that $\alpha\in\mathcal{F}(M,\xi)$ is non-degenerate, that is, all
the periodic orbits of the Reeb vector field $R_{\alpha}$ are non-degenerate.
Given an almost complex structure $J\in\mathcal{J}(\alpha)$ and $k$ points
$x_{1},\dots,x_{k}\in\widetilde{M}$, we define
$\mathcal{M}^{J}(x_{1},\dots,x_{k})$ to be the set of all pseudoholomorphic
curves $u:(\Sigma,j)\rightarrow(\widetilde{M},J)$ with the following
properties:

  1. 1.

The domain $(\Sigma,j)$ is a finite union of Riemann spheres with finitely
many punctures.

  2. 2.

The map $u$ is non-constant on every connected component of $\Sigma$.

  3. 3.

The points $x_{1},\dots,x_{k}$ are contained in the image of $u$.

  4. 4.

$u$ has finite energy.

  5. 5.

$u$ has at least one negative asymptotic orbit.

By Stokes’ theorem and exactness of $\widetilde{M}$, any curve
$u\in\mathcal{M}^{J}(x_{1},\dots,x_{k})$ has at least one positive asymptotic
orbit. We recall that the positive action $\mathcal{A}^{+}_{\alpha}(u)$ is the
sum of the $\alpha$-actions of these orbits. Given a non-degenerate
$\alpha\in\mathcal{F}(M,\xi)$, we define

$c_{k}(\alpha)\coloneqq\sup_{\begin{subarray}{c}J\in\mathcal{J}(\alpha)\\\
x_{1},\dots,x_{k}\in\widetilde{M}\end{subarray}}\inf_{u\in\mathcal{M}^{J}(x_{1},\dots,x_{k})}\mathcal{A}^{+}_{\alpha}(u)\in(0,+\infty].$
(10.1)

The proof of the basic properties of the functions $c_{k}$ which are listed in
the next theorem is similar to the analogous proofs in [MS23, Hut22a, Hut22b,
Edt22b].

######  Theorem 10.1.

The functions given by (10.1) have unique extensions
$c_{k}\colon\mathcal{F}(M,\xi)\rightarrow[0,+\infty]$, $k\geq 0$, satisfying
the following properties:

  1. 1.

(Scaling) $c_{k}(r\cdot\alpha)=r\cdot c_{k}(\alpha)$ for all positive real
numbers $r$.

  2. 2.

(Increasing) $c_{k}(\alpha)\leq c_{\ell}(\alpha)$ for $k\leq\ell$.

  3. 3.

(Sublinearity) $c_{k+\ell}(\alpha)\leq c_{k}(\alpha)+c_{\ell}(\alpha)$.

  4. 4.

(Invariance) $c_{k}(\varphi^{*}\alpha)=c_{k}(\alpha)$ for all
$\varphi\in\operatorname{Cont}(M,\xi)$.

  5. 5.

(Monotonicity): $c_{k}(\alpha)\leq c_{k}(\beta)$ whenever $\alpha\leq\beta$.

  6. 6.

(Lipschitz): If $c_{k}$ is finite, then $c_{k}$ is Lipschitz for the
$C^{0}$-norm on $\mathcal{F}(M,\xi)$.

  7. 7.

(Spectrality): If $c_{k}(\alpha)$ is finite, then it is a finite sum of
actions of periodic Reeb orbits of $\alpha$.

  8. 8.

(Packing): Suppose that $\alpha<\beta$ and assume that the ball $B^{2n}(A)$ of
symplectic width $A$ admits a symplectic embedding into
$\widetilde{M}^{\beta}_{\alpha}$. Then

$c_{k+1}(\beta)\geq c_{k}(\alpha)+A.$

######  Remark 10.2.

It follows from the monotonicity and scaling properties that, for fixed
$(M,\xi)$ and $k$, the spectral invariant $c_{k}$ is either finite on all of
$\mathcal{F}(M,\xi)$ or identically equal to $+\infty$. If $c_{k}$ is finite,
then the Weinstein conjecture holds for $(M,\xi)$.

###### Proof.

It is enough to check the above properties for non-degenerate contact forms.
Then, the Lipschitz property allows us to extend $c_{k}$ continuously to all,
possibly degenerate, contact forms and, by continuity, all properties of
$c_{k}$ continue to hold in the degenerate case.

Let $f_{r}$ be the diffeomorphism of $\mathbb{R}\times M$ given by
$f_{r}(a,p)=(ra,p)$. The pull-back by $f_{r}$ yields a bijection

$f_{r}^{*}:\mathcal{J}(r\alpha)\rightarrow\mathcal{J}(\alpha)$

and, for every $J\in\mathcal{J}(r\alpha)$ and every
$x_{1},\dots,x_{k}\in\widetilde{M}$, a bijection

$f_{r}^{*}:\mathcal{M}^{J}(x_{1},\dots,x_{k})\rightarrow\mathcal{M}^{f_{r}^{*}J}(f_{r}^{-1}(x_{1}),\dots,f_{r}^{-1}(x_{k})).$

For every $u\in\mathcal{M}^{J}(x_{1},\dots,x_{k})$, the positive action of $u$
with respect to $r\alpha$ is related to the positive action of $f_{r}^{*}(u)$
with respect to $\alpha$ by
$\mathcal{A}^{+}_{r\alpha}(u)=r\mathcal{A}^{+}_{\alpha}(f_{r}^{*}u)$. This
implies the scaling property $c_{k}(r\alpha)=rc_{k}(\alpha)$.

It is clear from the definition that $c_{k}$ is increasing in $k$ since adding
point constraints can only make the set of relevant curves smaller.

Sublinearity is also clear because the union of two curves with $k$ and $\ell$
point constraints, respectively, yields a curve with $k+\ell$ point
constraints.

Invariance follows because a contactomorphism
$\varphi\in\operatorname{Cont}(M,\xi)$ lifts to a symplectomorphism of the
symplectization which induces bijections between the relevant sets of
auxiliary data and curves.

The Lipschitz property follows from scaling and monotonicity.

Spectrality follows from the fact that the action spectrum of $\alpha$ is a
closed subset of $\mathbb{R}$.

The proofs of the monotonicity and packing property are more substantial and
rely on a neck-stretching argument and the SFT compactness theorem. Compared
to analogous arguments given in [Hut22a, Edt22b], the proofs are slightly
simplified because we have an apriori genus bound and can apply the SFT
compactness theorem directly. As these two proofs are very similar, we just
sketch the proof of the packing property.

We fix $\alpha<\beta$ and a symplectically embedded ball $B=B^{2n}(A)$ in
$\widetilde{M}_{\alpha}^{\beta}$. Moreover, we fix an almost complex structure
$J\in\mathcal{J}(\alpha)$ and $k$ points $x_{1},\dots,x_{k}\in\widetilde{M}$.
Let $\epsilon>0$ be arbitrarily small. Our goal is to find a $J$-holomorphic
curve $u\in\mathcal{M}^{J}(x_{1},\dots,x_{k})$ satisfying

$\mathcal{A}^{+}_{\alpha}(u)+A\leq c_{k+1}(\beta)+\epsilon.$

Since the constant $\epsilon$, the almost complex structure $J$, and the point
constraints are arbitrary, this implies the desired packing inequality.

Note that if we translate the almost complex structure $J$ and the points
$x_{i}$ using the canonical $\mathbb{R}$-action on $\widetilde{M}$, the new
moduli space $\mathcal{M}^{J}(x_{1},\dots,x_{k})$ is obtained from the old one
by applying the same translation. The positive action
$\mathcal{A}_{\alpha}^{+}(u)$ is not affected by translation. After
translating $J$ and the points $x_{i}$ sufficiently far downwards, we can
therefore assume w.l.o.g. that $J$ is adjusted to $\alpha$ on an open
neighbourhood of $\widetilde{M}_{\alpha}^{+}$ and that all points $x_{i}$ lie
below the hypersurface $M_{\alpha}$.

Fix a real number $S$ such that $J$ is adjusted to $\alpha$ on an open
neighbourhood of $\widetilde{M}_{e^{S}\alpha}^{-}$ and such that all $x_{i}$
lie above the hypersurface $M_{e^{S}\alpha}$. Let $T\in\mathbb{R}$ be such
that the hypersurface $M_{e^{T}\beta}$ lies below the hypersurface
$M_{e^{S}\alpha}$. Let $J_{\alpha}^{+}$ denote the $\alpha$-adjusted almost
complex structure whose restriction to $\widetilde{M}_{\alpha}^{+}$ agrees
with $J$ and let $J_{\alpha}^{-}$ denote the $\alpha$-adjusted almost complex
structure whose restriction to $\widetilde{M}_{e^{s}\alpha}^{-}$ agrees with
$J$.

Let $J_{\alpha}^{\beta}$ be a compatible almost complex structure which agrees
with $J_{\alpha}^{+}$ on a neighbourhood of $\widetilde{M}_{\alpha}^{-}$ and
which is equal to a $\beta$-adjusted almost complex structure $J_{\beta}^{+}$
on $\widetilde{M}_{\beta}^{+}$. Moreover, we assume that the restriction of
$J_{\alpha}^{\beta}$ to the ball $B$ agrees with the standard integrable
almost complex structure on Euclidean space.

Let $J_{\beta}^{\alpha}$ be a compatible almost complex structure agreeing
with $J_{\alpha}^{-}$ on a neighbourhood of $\widetilde{M}_{e^{S}\alpha}^{+}$
and whose restriction to $\widetilde{M}_{e^{T}\beta}^{-}$ agrees with a
$\beta$-adjusted almost complex structure $J_{\beta}^{-}$. Let us fix small
tubular neighbourhoods $\operatorname{nb}(M_{\alpha})$ and
$\operatorname{nb}(M_{e^{S}\alpha})$ and sequences of diffeomorphisms

$\varphi_{\nu}^{+}:\widetilde{M}_{e^{-\nu}\alpha}^{e^{\nu}\alpha}\rightarrow\operatorname{nb}(M_{\alpha}),\qquad\varphi_{\nu}^{-}:\widetilde{M}_{e^{-\nu}\alpha}^{e^{\nu}\alpha}\rightarrow\operatorname{nb}(M_{e^{S}\alpha})$

with the following properties:

  1. 1.

$\varphi_{\nu}^{\pm}$ maps translates of the hypersurface $M_{\alpha}$ to
translates of $M_{\alpha}$.

  2. 2.

$\pi\circ\varphi_{\nu}^{\pm}=\pi$ where $\pi:\widetilde{M}\rightarrow M$ is
the canonical projection.

  3. 3.

$d\varphi_{\nu}^{\pm}\partial_{a}=\partial_{a}$ near the boundary of
$\widetilde{M}_{e^{-\nu}\alpha}^{e^{\nu}\alpha}$.

Let us now define a sequence of almost complex structures
$J^{\nu}\in\mathcal{J}(\beta)$ as follows:

  1. 1.

$J^{\nu}$ agrees with $J_{\alpha}^{\beta}$ above
$\operatorname{nb}(M_{\alpha})$.

  2. 2.

$J^{\nu}$ agrees with $(\varphi_{\nu}^{+})_{*}J_{\alpha}^{+}$ on
$\operatorname{nb}(M_{\alpha})$.

  3. 3.

$J^{\nu}$ agrees with $J$ between $\operatorname{nb}(M_{\alpha})$ and
$\operatorname{nb}(M_{e^{S}\alpha})$.

  4. 4.

$J^{\nu}$ agrees with $(\varphi_{\nu}^{-})_{*}J_{\alpha}^{-}$ on
$\operatorname{nb}(M_{e^{S}\alpha})$.

  5. 5.

$J^{\nu}$ agrees with $J_{\beta}^{\alpha}$ below
$\operatorname{nb}(M_{e^{S}\alpha})$.

Let $x_{k+1}$ denote the center of $B$. By the definition of $c_{k+1}(\beta)$
given in (10.1), there exists a sequence of $J^{\nu}$-holomorphic curves
$u^{\nu}\in\mathcal{M}^{J^{\nu}}(x_{1},\dots,x_{k+1})$ such that

$\mathcal{A}^{+}_{\beta}(u^{\nu})\leq c_{k+1}(\beta)+\epsilon.$

The SFT compactness theorem implies that, after possibly passing to a
subsequence, the sequence of curves $u^{\nu}$ converges to a pseudo-
holomorphic building $\underline{u}=(u_{1},\dots,u_{N})$. Each $u_{i}$
consists of finitely many pseudoholomorphic nodal punctured finite-energy
spheres. In the following we will ignore nodal points and ghost bubbles and
regard each $u_{i}$ as a finite union of non-constant pseudoholomorphic
punctured finite-energy spheres. There exist indices $1\leq r<s<t\leq N$ such
that

  1. 1.

$u_{i}$ is $J_{\beta}^{+}$-holomorphic and positively and negatively
asymptotic to closed orbits of $R_{\beta}$ for all $1\leq i<r$.

  2. 2.

$u_{r}$ is $J_{\alpha}^{\beta}$-holomorphic and passes through $x_{k+1}$. It
is positively asymptotic to closed orbits of $R_{\beta}$ and negatively
asymptotic to closed orbits of $R_{\alpha}$.

  3. 3.

$u_{i}$ is $J_{\alpha}^{+}$-holomorphic and positively and negatively
asymptotic to closed orbits of $R_{\alpha}$ for all $r<i<s$.

  4. 4.

$u_{s}$ is $J$-holomorphic and passes through $x_{1},\dots,x_{k}$. It is
positively and negatively asymptotic to closed orbits of $R_{\alpha}$.

  5. 5.

$u_{i}$ is $J_{\alpha}^{-}$-holomorphic and positively and negatively
asymptotic to closed orbits of $R_{\alpha}$ for all $s<i<t$.

  6. 6.

$u_{t}$ is $J_{\beta}^{\alpha}$-holomorphic. It is positively asymptotic to
closed orbits of $R_{\alpha}$ and negatively asymptotic to closed orbits of
$R_{\beta}$.

  7. 7.

$u_{i}$ is $J_{\beta}^{-}$-holomorphic and positively and negatively
asymptotic to closed orbits of $R_{\beta}$ for all $t<i\leq N$.

Note that since every $u^{\nu}$ has at least one negative end, each $u_{i}$
must have at least one negative end. Therefore,
$u_{s}\in\mathcal{M}^{J}(x_{1},\dots,x_{k})$. We need to estimate the positive
action $\mathcal{A}^{+}_{\alpha}(u_{s})$. For every $i$, the negative
asymptotic orbits of $u_{i}$ match the positive asymptotic orbits of
$u_{i+1}$. This implies that

$\sum\limits_{i=1}^{r-1}\mathcal{A}_{\beta}(u_{i})+\mathcal{A}_{\alpha}^{\beta}(u_{r})+\sum\limits_{i=r+1}^{s-1}\mathcal{A}_{\alpha}(u_{i})+\mathcal{A}_{\alpha}^{+}(u_{s})=\mathcal{A}_{\beta}^{+}(u_{1}).$

Since $\mathcal{A}_{\beta}^{+}(u^{\nu})\leq c_{k+1}(\beta)+\varepsilon$ for
all $\nu$ and the positive ends of $u^{\nu}$ are converging to the positive
ends of $u_{1}$, we have

$\mathcal{A}_{\beta}^{+}(u_{1})\leq c_{k+1}(\beta)+\varepsilon.$

The actions $\mathcal{A}_{\beta}(u_{i})$ are non-negative for $1\leq i<r$.
Similarly, the actions $\mathcal{A}_{\alpha}(u_{i})$ are non-negative for
$r<i<s$. Since $u_{r}$ passes through the center of the ball $B$ equipped with
the standard complex structure, $\mathcal{A}_{\alpha}^{\beta}(u_{r})\geq A$ by
the monotonicity lemma. Combining the above estimates yields

$\mathcal{A}_{\alpha}^{+}(u_{s})+A\leq c_{k+1}(\beta)+\varepsilon$

as desired. ∎

In the next result, we determine the value of the first two spectral
invariants for Zoll contact forms.

######  Proposition 10.3.

Let $\alpha_{0}\in\mathcal{F}(M,\xi)$ be a Zoll contact form with minimal
period $\operatorname{sys}(\alpha_{0})$. Then we have

$c_{0}(\alpha_{0})=c_{1}(\alpha_{0})=\operatorname{sys}(\alpha_{0}).$

###### Proof.

Let $\alpha$ be a $C^{\infty}$-small non-degenerate perturbation of
$\alpha_{0}$. Fix an almost complex structure $J\in\mathcal{J}(\alpha)$ and a
point $x\in\widetilde{M}$. Our goal is to show that there exists a
$J$-holomorphic curve $u\in\mathcal{M}^{J}(x)$ with positive action bounded
from above by

$\mathcal{A}^{+}_{\alpha}(u)\leq\tfrac{3}{2}\operatorname{sys}(\alpha_{0}).$

Since $J$ and $x$ are arbitrary, this implies that
$c_{1}(\alpha)\leq\frac{3}{2}\operatorname{sys}(\alpha_{0})$. It then follows
from continuity of $c_{1}$ that
$c_{1}(\alpha_{0})\leq\frac{3}{2}\operatorname{sys}(\alpha_{0})$. The
increasing and the spectrality properties imply that we must have
$c_{0}(\alpha_{0})=c_{1}(\alpha_{0})=\operatorname{sys}(\alpha_{0})$.

We construct the pseudoholomorphic curve $u$ by neck-stretching
pseudoholomorphic spheres in a ruled symplectic manifold $(X,\omega)$ which
contains $M_{\alpha}$ as a contact type hypersurface. The symplectic manifold
$(X,\omega)$ is constructed by performing a so-called symplectic cut, see
[Ler95], as follows. Identify the symplectization $\widetilde{M}$ with
$\mathbb{R}\times M$ by means of the diffeomorphism $\varphi_{\alpha_{0}}$, so
that $\widetilde{\omega}$ is given by $d(e^{s}\alpha_{0})$. Fix a constant
$S>0$. The boundary components of the compact region $[-S,S]\times M$ in the
symplectization are foliated by closed characteristic leaves. The symplectic
manifold $(X,\omega)$ is obtained from $[-S,S]\times M$ by collapsing each of
these leaves to a point. Let $B$ denote the quotient of $M$ by the free
$S^{1}$-action induced by the Zoll Reeb flow. The space $X$ naturally fibers
over $B$. The fibers are symplectic $2$-spheres of the form
$([-S,S]\times\gamma)/\sim$ where $\gamma$ is a closed Reeb orbit of
$\alpha_{0}$ and $\sim$ collapses the circles $\left\\{\pm
S\right\\}\times\gamma$ to points. Under the quotient, the two boundary
components of $[-S,S]\times M$ yield smooth symplectic divisors $B_{\pm}$ of
$X$ both diffeomorphic to $B$. Clearly, $X$ contains $M_{\alpha}$ as a contact
type hypersurface. The function
$(s,p)\mapsto-\operatorname{sys}(\alpha_{0})e^{s}$ on $\mathbb{R}\times M$
induces a smooth Hamiltonian $H$ on $X$ which generates a Hamiltonian
$S^{1}$-action. The two divisors $B_{\pm}$ are fixed point sets of this action
and the restriction of the action to the complement $X\setminus(B_{+}\cup
B_{-})$ is free. In particular, the assumptions in [McD09, Proposition 4.3]
are satisfied. Let $F\in H_{2}(X;\mathbb{Z})$ denote the homology class of a
fiber. Then [McD09, Proposition 4.3] implies that the Gromov–Witten invariant
$\langle\mathrm{pt}\rangle_{F}^{X}$ does not vanish. We conclude that, for
every $\omega$-compatible almost complex structure $\widetilde{J}$ on $X$ and
for every point $p\in X$, there exists a nodal $\widetilde{J}$-holomorphic
sphere passing through $p$ and representing the homology class $F$.

Let us now choose a constant $S^{\prime}$ large enough such that the almost
complex structure $J\in\mathcal{J}(\alpha)$ fixed at the beginning of the
proof is adjusted to $\alpha$ on neighbourhoods of the sets
$\widetilde{M}_{e^{\pm S^{\prime}}\alpha}^{\pm}$. Moreover, we assume that the
point $x$ strictly lies between the two hypersurfaces $M_{e^{\pm
S^{\prime}}\alpha}$. We set the constant $S$ appearing in the construction of
$X$ to be $S\coloneqq S^{\prime}+\ln(3/2)$. Since $\alpha$ is
$C^{\infty}$-close to $\alpha_{0}$, the hypersurfaces $M_{e^{\pm
S^{\prime}}\alpha}$ lie between the hypersurfaces $M_{e^{\pm S}\alpha_{0}}$
and do not intersect them.

We define a sequence of compatible almost complex structures $J^{\nu}$ on $X$
as follows. Let $J_{\alpha}^{\pm}$ denote the $\alpha$-adjusted almost complex
structure whose restriction to $\widetilde{M}_{e^{\pm S^{\prime}}}^{\pm}$
agrees with $J$. Let $X_{\pm}\subset X$ denote the connected component of
$X\setminus M_{e^{\pm S^{\prime}}\alpha}$ containing $B_{\pm}$. Let
$\widehat{X}_{\pm}$ denote the symplectic completion of $X_{\pm}$ obtained by
attaching the cylindrical end $\widetilde{M}_{e^{\pm
S^{\prime}}\alpha}^{\mp}$. Let $J_{\pm}$ be a compatible almost complex
structure on $\widehat{X}_{\pm}$ agreeing with $J_{\alpha}^{\pm}$ in a
neighbourhood of the cylindrical end $\widetilde{M}_{e^{\pm
S^{\prime}}\alpha}^{\mp}$.

Let us fix small tubular neighbourhoods $\operatorname{nb}(M_{e^{\pm
S^{\prime}}\alpha})$ and sequences of diffeomorphisms

$\varphi_{\nu}^{\pm}:\widetilde{M}_{e^{-\nu}\alpha}^{e^{\nu}\alpha}\rightarrow\operatorname{nb}(M_{e^{\pm
S^{\prime}}\alpha})$

with the following properties:

  1. 1.

$\varphi_{\nu}^{\pm}$ maps translates of the hypersurface $M_{\alpha}$ to
translates of $M_{\alpha}$.

  2. 2.

$\pi\circ\varphi_{\nu}^{\pm}=\pi$ where $\pi:\widetilde{M}\rightarrow M$ is
the canonical projection.

  3. 3.

$d\varphi_{\nu}^{\pm}\partial_{a}=\partial_{a}$ near the boundary of
$\widetilde{M}_{e^{-\nu}\alpha}^{e^{\nu}\alpha}$.

Let us now define the compatible almost complex structure $J^{\nu}$ on $X$ as
follows:

  1. 1.

$J^{\nu}$ agrees with $J_{+}$ above
$\operatorname{nb}(M_{e^{S^{\prime}}\alpha})$.

  2. 2.

$J^{\nu}$ agrees with $(\varphi_{\nu}^{+})_{*}J_{\alpha}^{+}$ on
$\operatorname{nb}(M_{e^{S^{\prime}}\alpha})$.

  3. 3.

$J^{\nu}$ agrees with $J$ between
$\operatorname{nb}(M_{e^{S^{\prime}}\alpha})$ and
$\operatorname{nb}(M_{e^{-S^{\prime}}\alpha})$.

  4. 4.

$J^{\nu}$ agrees with $(\varphi_{\nu}^{-})_{*}J_{\alpha}^{-}$ on
$\operatorname{nb}(M_{e^{-S^{\prime}}\alpha})$.

  5. 5.

$J^{\nu}$ agrees with $J_{-}$ below
$\operatorname{nb}(M_{e^{-S^{\prime}}\alpha})$.

By the above discussion, we may choose, for every $\nu$, a
$J^{\nu}$-holomorphic sphere $u^{\nu}$ representing the homology class of the
fiber $F$ and passing through the point $x\in X$. By the SFT compactness
theorem, this sequence converges, after possibly passing to a subsequence, to
a pseudoholomorphic building $\underline{u}=(u_{1},\dots,u_{N})$. Every
$u_{i}$ consists of finitely many nodal punctured finite-energy
pseudoholomorphic spheres. As before, we ignore nodes and ghost bubbles and
regard each $u_{i}$ as a finite union of punctured finite energy spheres.

The homological intersection number $F\cdot B_{\pm}$ is equal to $1$. This
implies that every $u^{\nu}$ intersects both $B_{\pm}$. Therefore, $u_{1}$ is
a $J_{+}$-holomorphic curve in $\widehat{X}_{+}$ and its intersection number
with $B_{+}$ is equal to $1$. Similarly, $u_{N}$ is a $J_{-}$-holomorphic
curve in $\widehat{X}_{-}$ intersecting $B_{-}$ with intersection number $1$.
Since the negative asymptotic limit orbits of $u_{i}$ match the positive
asymptotic limit orbits of $u_{i+1}$, we conclude that every $u_{i}$ for
$1<i<N$ must have both positive and negative asymptotic orbits. Let $1<r<N$ be
the index such that $u_{r}$ is a $J$-holomorphic curve in $\widetilde{M}$
passing through $x\in\widetilde{M}$. Since $u_{r}$ has a negative asymptotic
orbit, we have $u_{r}\in\mathcal{M}^{J}(x)$. It remains to estimate
$\mathcal{A}_{\alpha}^{+}(u)$.

The complement of $B_{+}$ in $\widehat{X}_{+}$ is exact. Indeed, restricting
the natural Liouville $1$-form $\widetilde{\lambda}$ on $\widetilde{M}$ to the
interior of $\widetilde{M}_{e^{S}\alpha_{0}}^{-}$ yields a primitive $\lambda$
of the symplectic form $\omega_{+}$ on $\widehat{X}_{+}\setminus B_{+}$.
Clearly, the primitive $\lambda$ does not extend over $B_{+}$. In fact, if
$\Sigma$ is a compact oriented surface, possibly with boundary, and
$v:\Sigma\rightarrow\widehat{X}_{+}$ is a smooth map sending $\partial\Sigma$
into the complement of $B_{+}$, then

$\int_{\Sigma}v^{*}\omega_{+}=\int_{\partial\Sigma}v^{*}\lambda+e^{S}\operatorname{sys}(\alpha_{0})v\cdot
B_{+}.$

In order to make sense of the homological intersection numer $v\cdot B_{+}$,
we regard $v$ as a $2$-chain with fixed boundary in the complement of $B_{+}$.
Applying this identity to $u_{1}$ yields

$\int_{u_{1}^{-1}(\widetilde{M}^{-}_{e^{S^{\prime}}\alpha})}(\varphi_{\alpha}^{-1}\circ
u_{1})^{*}d\alpha+\int_{u_{1}^{-1}(X_{+})}u_{1}^{*}\omega_{+}=-\mathcal{A}_{e^{S^{\prime}}\alpha}^{-}(u_{1})+e^{S}\operatorname{sys}(\alpha_{0}).$

Since $u_{1}$ is pseudoholomorphic, the left-hand side of this identity is
non-negative. We deduce that

$e^{S^{\prime}}\mathcal{A}_{\alpha}^{-}(u_{1})=\mathcal{A}_{e^{S^{\prime}}\alpha}^{-}(u_{1})\leq
e^{S}\operatorname{sys}(\alpha_{0}).$ (10.2)

It follows from our choice $S=S^{\prime}+\ln(3/2)$ that
$\mathcal{A}_{\alpha}^{-}(u_{1})\leq\frac{3}{2}\operatorname{sys}(\alpha_{0})$.

The curve $u_{i}$ is a $J_{\alpha}^{+}$-holomorphic curve in $\widetilde{M}$
for $1<i<r$. We have

$\sum\limits_{i=2}^{r-1}\mathcal{A}_{\alpha}(u_{i})+\mathcal{A}_{\alpha}^{+}(u_{r})=\mathcal{A}_{\alpha}^{-}(u_{1}).$

Since the actions $\mathcal{A}_{\alpha}(u_{i})$ are non-negative for $1<i<r$,
we deduce that

$\mathcal{A}_{\alpha}^{+}(u_{r})\leq\mathcal{A}_{\alpha}^{-}(u_{1})\leq\frac{3}{2}\operatorname{sys}(\alpha_{0}).$

This concludes the proof of the proposition. ∎

Next, we compute the first two spectral invariants for contact forms which are
$C^{2}$-small, $S^{1}$-invariant perturbations of a Zoll one.

######  Proposition 10.4.

Let $\alpha_{0}\in\mathcal{F}(M,\xi)$ be a Zoll contact form with minimal
period $\mathrm{sys}(\alpha_{0})$. If the smooth function
$S:M\rightarrow\mathbb{R}$ is $C^{2}$-close to the constant function 1 and
$S^{1}$-invariant, then

$c_{0}(S\alpha_{0})=\mathrm{sys}(\alpha_{0})\min_{M}S,\qquad
c_{1}(S\alpha_{0})=\mathrm{sys}(\alpha_{0})\max_{M}S.$

###### Proof.

We abbreviate $\max S\coloneqq\max_{M}S$ and $\min S\coloneqq\min_{M}S$. Our
proof is based on the following claims.

Claim 1. If $S$ is $C^{2}$-close to 1 and the positive number
$\overline{S}>\max S$ is close to $\max S$, then there is a symplectic
embedding of the $2n$-dimensional ball of symplectic width
$\mathrm{sys}(\alpha_{0})(\max S-\min S)$ into the cobordism
$\widetilde{M}_{S\alpha_{0}}^{\overline{S}\alpha_{0}}$.

Claim 2. If $S$ is $C^{2}$-close to 1 and the positive number
$\underline{S}<\min S$ is close to $\min S$, then there is a symplectic
embedding of the $2n$-dimensional ball of symplectic width
$\mathrm{sys}(\alpha_{0})(\max S-\min S)$ into the cobordism
$\widetilde{M}_{\underline{S}\alpha_{0}}^{S\alpha_{0}}$.

Before proving these claims, we show how they allow us to determine
$c_{0}(S\alpha_{0})$ and $c_{1}(S\alpha_{0})$. By Claim 1, the packing
property applied to $\widetilde{M}_{S\alpha_{0}}^{\overline{S}\alpha_{0}}$,
the scaling property and Proposition 10.3, we have

$c_{0}(S\alpha_{0})+\mathrm{sys}(\alpha_{0})(\max S-\min S)\leq
c_{1}(\overline{S}\alpha_{0})=\overline{S}\,c_{1}(\alpha_{0})=\overline{S}\,\mathrm{sys}(\alpha_{0}),$

and hence, letting $\overline{S}$ tend to $\max S$, we get

$c_{0}(S\alpha_{0})\leq\mathrm{sys}(\alpha_{0})\min S.$

Since the right-hand side is the minimum of the action spectrum of
$S\alpha_{0}$, the spectrality property implies that the above inequality is
an equality. Similarly, the packing property applied to
$\widetilde{M}_{\underline{S}\alpha_{0}}^{S\alpha_{0}}$, the scaling property
and Proposition 10.3 imply

$c_{1}(S{\alpha_{0}})\geq
c_{0}(\underline{S}\alpha_{0})+\mathrm{sys}(\alpha_{0})(\max S-\min
S)=\underline{S}\,c_{0}(\alpha_{0})+\mathrm{sys}(\alpha_{0})(\max S-\min S).$

Letting $\underline{S}$ tend to $\min S$, we get
$c_{1}(S\alpha_{0})\geq\mathrm{sys}(\alpha_{0})\max S$. By monotonicity, we
have

$c_{1}(S\alpha_{0})\leq c_{1}((\max
S)\,\alpha_{0})=\mathrm{sys}(\alpha_{0})\max S,$

and the desired formula for $c_{1}(S\alpha_{0})$ follows.

There remains to prove the above claims. We prove Claim 1, the proof of Claim
2 being analogous. Up to rescaling, we may assume that
$\mathrm{sys}(\alpha_{0})=1$. We identify the symplectization $\widetilde{M}$
with $\mathbb{R}\times M$ via $\varphi_{\alpha_{0}}$, so that
$\widetilde{\omega}=d(e^{a}\alpha_{0})$ and

$M_{S\alpha_{0}}^{\overline{S}\alpha_{0}}=\\{(a,q)\in\mathbb{R}\times M\mid
S(q)<e^{a}<\overline{S}\\}.$

Denote by $p:M\rightarrow Y$ the $\mathbb{T}$-bundle,
$\mathbb{T}\coloneqq\mathbb{R}/\mathbb{Z}$, whose base is the quotient of $M$
by the free action of the Reeb flow of $\alpha_{0}$. Then $Y$ is a
$(2n-2)$-dimensional manifold with a symplectic form $\omega$ satisfying
$p^{*}\omega=d\alpha_{0}$. By means of a Darboux trivializing chart, we can
identify a neighborhood $(U_{y},\omega)$ of any $y$ in $Y$ with a ball
$\widehat{B}_{\rho}$ of radius $\rho>0$ in
$(\mathbb{C}^{n-1},\widetilde{\omega}_{0})$, where $\rho$ is independent of
$y$. Moreover, we can identify $p^{-1}(U_{y})$ with
$\mathbb{T}\times\widehat{B}_{\rho}$ in such a way that $\alpha_{0}$
corresponds to $\widehat{\lambda}_{0}+dt$. Here, $t$ denotes the variable in
$\mathbb{T}$ and $\widehat{\lambda}_{0}$ is the standard primitive of the
standard symplectic form $\widehat{\omega}_{0}$ on $\mathbb{C}^{n-1}$ (we are
using the notation from Section 1).

Correspondingly, we obtain an identification of the open subset
$\widetilde{U}_{y}\coloneqq\mathbb{R}\times p^{-1}(U_{y})$ of $\widetilde{M}$
with $\mathbb{R}\times\mathbb{T}\times\widehat{B}_{\rho}$, with respect to
which the 1-form $e^{a}\alpha_{0}$ reads $e^{a}(dt+\widehat{\lambda}_{0})$.
The diffeomorphism

$\Psi_{y}:\\{(s,t,w)\in(0,+\infty)\times\mathbb{T}\times\mathbb{C}^{n-1}\mid|w|<s^{\frac{1}{2}}\rho\\}\rightarrow\mathbb{R}\times\mathbb{T}\times\widehat{B}_{\rho}=\widetilde{U}_{y}\subset\widetilde{M}$

given by $\Psi_{y}(s,t,w)\coloneqq(\log s,t,s^{-\frac{1}{2}}w)$ satisfies

$\Psi_{y}^{*}(e^{a}\alpha_{0})=\Psi_{y}^{*}(e^{a}(dt+\widehat{\lambda}_{0}))=s\,dt+\widehat{\lambda}_{0},$

and hence

$\Psi_{y}^{*}\widetilde{\omega}=\Psi_{y}^{*}d(e^{a}\alpha_{0})=ds\wedge
dt+\widetilde{\omega}_{0}.$

Now we choose $y\in Y$ so that the $\mathbb{T}$-invariant function $S$
achieves its minimum at the circle $p^{-1}(y)$. In the above identifications,
we then have $S(t,0)=\min S$ for every $t\in\mathbb{T}$,

$\widetilde{M}_{S\alpha_{0}}^{\overline{S}\alpha_{0}}\cap\widetilde{U}_{y}=\\{(a,t,u)\in\mathbb{R}\times\mathbb{T}\times\widehat{B}_{\rho}\mid
S(t,u)<e^{a}<\overline{S}\\},$

and hence it is enough to symplectically embed a $2n$-ball of symplectic width
$\max S-\min S$ into the open subset

$V\coloneqq\Psi^{-1}_{y}(\widetilde{M}_{S\alpha_{0}}^{\overline{S}\alpha_{0}}\cap\widetilde{U}_{y})=\\{(s,t,w)\in(0,+\infty)\times\mathbb{T}\times\mathbb{C}^{n-1}\mid
S(t,s^{-\frac{1}{2}}w)<s<\overline{S},\;|w|<s^{\frac{1}{2}}\rho\\}$

of $((0,+\infty)\times\mathbb{T}\times\mathbb{C}^{n-1},ds\wedge
dt+\widehat{\omega}_{0})$. Since $S(t,0)=\min S$, we have

$S(t,u)\leq\min
S+c\|\nabla^{2}S\|_{C^{0}}|u|^{2}\qquad\forall(t,u)\in\mathbb{T}\times\widehat{B}_{\rho},$

for a suitable constant $c>0$. By this bound, the inequality
$s>S(t,s^{-\frac{1}{2}}w)$ appearing in the representation of $V$ is implied
by

$s>\min S+c\|\nabla^{2}S\|_{C^{0}}\frac{|w|^{2}}{s},$

which is equivalent to

$s>\frac{1}{2}\min S\left(1+\sqrt{1+\frac{4c\|\nabla^{2}S\|_{C^{0}}}{(\min
S)^{2}}|w|^{2}}\right).$

Since $\sqrt{1+r}\leq 1+\frac{r}{2}$ for every $r\geq 0$, the above inequality
is implied by

$s>\min S+\frac{c}{\min S}\|\nabla^{2}S\|_{C^{0}}|w|^{2},$

and we conclude that $V$ contains the set

$V^{\prime}\coloneqq\\{(s,t,w)\in(0,+\infty)\times\mathbb{T}\times\mathbb{C}^{n-1}\mid
f(|w|^{2})<s<\overline{S}\\},$

where

$f(r)\coloneqq\max\left\\{\min S+\frac{c\|\nabla^{2}S\|_{C^{0}}}{\min
S}r,\frac{r}{\rho^{2}}\right\\}.$

If $S$ is $C^{2}$-close to $1$ and $\overline{S}$ is close to $\max S$, then
both $\overline{S}-\min S$ and $\|\nabla^{2}S\|_{C^{0}}$ are small and we can
guarantee that

$f(r)\leq\min S+\pi r\qquad\forall
r\in\bigl{[}0,{\textstyle\frac{1}{\pi}}(\overline{S}-\min S)\bigr{]}.$

This implies that $V^{\prime}$ contains the set

$V^{\prime\prime}\coloneqq\bigl{\\{}(s,t,w)\in(0,+\infty)\times\mathbb{T}\times\mathbb{C}^{n-1}\mid\min
S+\pi|w|^{2}<s<\overline{S}\bigr{\\}}.$

Let $\Phi:\Omega\rightarrow\mathbb{C}^{*}\times\mathbb{C}^{n-1}$ be the
symplectomorphism defined in (1.1). By (1.3), the symplectomorphism

$(s,t,w)\mapsto\Phi\bigr{(}s-\min S-\pi,t,w\bigr{)}$

maps $V^{\prime\prime}$ onto $B(\rho)\setminus\\{z_{1}=0\\}$, where $B(\rho)$
is the ball of $\mathbb{C}^{n}$ of symplectic width
$\rho\coloneqq\overline{S}-\min S$ and $\\{z_{1}=0\\}$ is a complex
hyperplane. By Traynor’s embedding construction (see Construction 3.2 in
[Tra95]), for every $\rho^{\prime}\in(0,\rho)$, there is a symplectic
embedding $\Psi\colon B(\rho^{\prime})\to B(\rho)\setminus\\{z_{1}=0\\}$. In
the simple case we are considering here, the embedding is constructed as
follows. First, let $\psi\colon D(\rho^{\prime})\to D(\rho)\setminus\\{0\\}$
be an area preserving embedding such that

$\pi|\psi(\zeta)|^{2}\leq\pi|\zeta|^{2}+(\rho-\rho^{\prime}),\qquad\forall\,\zeta\in
D(\rho^{\prime}).$

Then,

$\Psi(\zeta_{1},\ldots,\zeta_{n})\coloneqq(\psi(\zeta_{1}),\zeta_{2},\ldots,\zeta_{n}),\qquad(\zeta_{1},\ldots,\zeta_{n})\in
B(\rho^{\prime})$

is the required symplectic embedding. Choosing $\rho^{\prime}=\max S-\min
S<\overline{S}-\min S=\rho$, we deduce that $V\supset V^{\prime\prime}$
contains a symplectic ball of width $\max S-\min S$. ∎

##  11 Proof of Theorem 5

Let $\alpha\in\mathcal{F}(M,\xi)$ be $C^{2}$-close to the Zoll contact form
$\alpha_{0}\in\mathcal{F}(M,\xi)$. By Theorem 3, there is
$\varphi\in\mathrm{Cont}_{0}(M,\xi)$ such that

$\varphi^{*}\alpha=Te^{h}\alpha_{0},$

where the functions $T$ and $h$ satisfy the conditions (a)-(e) listed in that
theorem. We denote by $T_{\min}(\alpha)=\mathrm{sys}(\alpha)$ and
$T_{\max}(\alpha)$ the minimum and the maximum of the periods of the short
closed orbits of $R_{\alpha}$, which by (e) satisfy

$T_{\min}(\alpha)=\mathrm{sys}(\alpha_{0})\min T,\qquad
T_{\max}(\alpha)=\mathrm{sys}(\alpha_{0})\max T.$ (11.1)

Then $\varphi^{*}\alpha=e^{f}\alpha_{0}$ with

$f\coloneqq h+\log T,$

and by property (c) of Theorem 3 we have

$\max_{M}f-\min_{M}f=\log\frac{T_{\max}(\alpha)}{T_{\min}(\alpha)}.$

This shows that

$d(\alpha_{0},\alpha)\leq\log\frac{T_{\max}(\alpha)}{T_{\min}(\alpha)}.$

In order to conclude our proof, we need to prove the reverse inequality

$d(\alpha_{0},\alpha)\geq\log\frac{T_{\max}(\alpha)}{T_{\min}(\alpha)}.$
(11.2)

We shall deduce this from the following claim.

Claim. $c_{0}(\alpha)=T_{\min}(\alpha)$ and $c_{1}(\alpha)=T_{\max}(\alpha)$.

Postponing its proof, we show how this claim implies (11.2). Let
$\psi\in\mathrm{Cont}_{0}(M,\xi)$ and let $g$ be the smooth positive function
such that

$\psi^{*}\alpha=e^{g}\alpha_{0}.$

Denoting by

$a\coloneqq e^{\min_{M}g}=\min_{M}e^{g},\qquad b\coloneqq
e^{\max_{M}g}=\max_{M}e^{g},$

we have

$a\alpha_{0}\leq\psi^{*}\alpha\leq b\alpha_{0}.$

Using Proposition 10.3, the scaling, monotonicity and invariance properties of
the spectral invariants, and the above claim, we compute

$a\,\mathrm{sys}(\alpha_{0})=c_{0}(a\,\alpha_{0})\leq
c_{0}(\psi^{*}\alpha)=c_{0}(\alpha)=T_{\min}(\alpha),$

and

$b\,\mathrm{sys}(\alpha_{0})=c_{1}(b\,\alpha_{0})\geq
c_{1}(\psi^{*}\alpha)=c_{1}(\alpha)=T_{\max}(\alpha).$

Therefore

$\max_{M}g-\min_{M}g=\log\frac{b}{a}\geq\log\frac{T_{\max}(\alpha)}{T_{\min}(\alpha)}.$

Since the contactomorphism $\psi\in\mathrm{Cont}_{0}(M,\xi)$ was arbitrary,
the above inequality implies (11.2).

There remains to prove the claim. In the notation from the proof of Theorem 3,
we have

$v^{*}u^{*}\alpha=S\alpha_{0}+\eta.$

By invariance, this implies that

$c_{k}(\alpha)=c_{k}(S\alpha_{0}+\eta),$

where $c_{k}$ is viewed as a function on
$\mathcal{F}(M,\operatorname{ker}(S\alpha_{0}+\eta))$ on the right hand side.
For $t\in[0,1]$, we define

$\gamma_{t}\coloneqq S\alpha_{0}+t\eta.$

Since $S$ is $C^{1}$-close to $1$ and $\eta$ and $d\eta$ are both
$C^{0}$-small, $\gamma_{t}$ is a contact form for all $t$. Moreover,
$\gamma_{t}$ and $d\gamma_{t}$ are $C^{0}$-close to $\alpha_{0}$ and
$d\alpha_{0}$, respectively.

Claim. The short closed Reeb orbits of $\gamma_{t}$ and their periods are
independent of $t\in[0,1]$.

###### Proof.

Let us split the Reeb vector field $R_{\gamma_{t}}$ of $\gamma_{t}$ into a
component $X_{t}$ parallel to $R_{\alpha_{0}}$ and a component $Y_{t}$ tangent
to $\xi=\operatorname{ker}\alpha_{0}$. Since $\gamma_{t}$ and $d\gamma_{t}$
are $C^{0}$-close to $\alpha_{0}$ and $d\alpha_{0}$, the component $X_{t}$ is
$C^{0}$-close to $R_{\alpha_{0}}$ and the component $Y_{t}$ is $C^{0}$-small.
Let us write $X_{t}=a_{t}R_{\alpha_{0}}$ for some function $a_{t}$ which is
$C^{0}$-close to $1$. Then a direct computation shows that

$\iota_{X_{t}}d\gamma_{t}=a_{t}(-dS+tF[dS])$

and therefore

$\iota_{Y_{t}}d\gamma_{t}=a_{t}(dS-tF[dS]).$

Recall that $dS=-B[V]$ where
$B:\operatorname{ker}\alpha_{0}\rightarrow(\operatorname{ker}\alpha_{0})^{*}$
is $C^{0}$-close to the isomorphism induced by $d\alpha_{0}$ and $V$ is a
section of $\operatorname{ker}\alpha_{0}$ which is $C^{1}$-small. Let
$B_{t}:\operatorname{ker}\alpha_{0}\rightarrow(\operatorname{ker}\alpha_{0})^{*}$
denote the isomorphism induced by $d\gamma_{t}$, which is also $C^{0}$-close
to the isomorphism induced by $d\alpha_{0}$. We have

$Y_{t}=a_{t}B_{t}^{-1}(B[V]-tF[B[V]])=a_{t}Q_{t}[V]$

where $Q_{t}$ is a suitable isomorphism of $\operatorname{ker}\alpha_{0}$
which is $C^{0}$-close to the identity. We may therefore write
$R_{\gamma_{t}}=a_{t}(R_{\alpha_{0}}+Q_{t}[V])$. The argument in the proof of
Proposition 4.3 (ii) shows that the short closed Reeb orbits of $\gamma_{t}$
are precisely the zero set of $V$ or equivalently the critical set of $S$. The
action of such an orbit $\gamma$ is simply given by
$S(\gamma)\operatorname{sys}(\alpha_{0})$ and in particular independent of
$t$. ∎

The above claim shows that the short action spectrum of $\gamma_{t}$ is
independent of $t$. The function $t\mapsto c_{k}(\gamma_{t})$ for
$k\in\left\\{0,1\right\\}$ is continuous and takes values in this short action
spectrum. Since the action spectrum has measure zero, this implies that
$t\mapsto c_{k}(\gamma_{t})$ is constant for $k\in\left\\{0,1\right\\}$. In
particular, we obtain

$c_{0}(\alpha)=c_{0}(S\alpha_{0})\quad\text{and}\quad
c_{1}(\alpha)=c_{1}(S\alpha_{0}).$

Next, we define $\beta_{t}\coloneqq(1-t)S\alpha_{0}+t\alpha_{0}$. Since $S$ is
$C^{1}$-close to $1$, this is a contact form for all $t\in[0,1]$.

Claim. The short closed Reeb orbits of $\beta_{t}$ are independent of
$t\in(0,1]$. If $\gamma$ is such a short closed Reeb orbit, then its action
with respect to $\beta_{t}$ is given by
$((1-t)S(\gamma)+t)\operatorname{sys}(\alpha_{0})$.

###### Proof.

The proof of this claim is similar to the proof of the claim above. ∎

It follows from continuity of the spectral invariants, the above claim and the
fact that action spectra have measure zero that

$c_{k}(\beta_{t})=(1-t)c_{k}(S\alpha_{0})+t\operatorname{sys}(\alpha_{0})\quad\text{for
$k\in\left\\{0,1\right\\}$}.$

If $t>0$ is sufficiently small, then $(1-t)S+t$ is $C^{2}$-small. Therefore,
we may apply Proposition 10.4 to the contact form $\beta_{t}$ for $t>0$ small.
We conclude that

$c_{0}(\beta_{t})=\min_{M}((1-t)S+t)\operatorname{sys}(\alpha_{0})\quad\text{and}\quad
c_{1}(\beta_{t})=\max_{M}((1-t)S+t)\operatorname{sys}(\alpha_{0}).$

In combination with the above expression for $c_{k}(\beta_{t})$, this implies

$c_{0}(S\alpha_{0})=\min_{M}S\operatorname{sys}(\alpha_{0})=T_{\min}(\alpha)\quad\text{and}\quad
c_{1}(S\alpha_{0})=\max_{M}S\operatorname{sys}(\alpha_{0})=T_{\max}(\alpha).$

This concludes the proof of the claim.

##  Appendix A Appendix: Construction of exotic ellipsoids

### The statements.

Denote by $\alpha_{0}=\lambda_{0}|_{S^{3}}$ the standard contact form on the
3-sphere, see (6), by $\xi_{\mathrm{st}}=\ker\alpha_{0}$ the induced co-
oriented contact structure, and by $\mathcal{F}(S^{3},\xi_{\mathrm{st}})$ the
space of contact forms on $S^{3}$ which define $\xi_{\mathrm{st}}$. This space
can be identified with the space of positive smooth functions on $S^{3}$ and
is endowed with the $C^{\infty}$-topology. If $(a,b)$ is a pair of positive
real numbers, we denote by $E(a,b)$ the open ellipsoid

$E(a,b)\coloneqq\Bigr{\\{}(z_{1},z_{2})\in\mathbb{C}^{2}\mid\frac{\pi}{a}|z_{1}|^{2}+\frac{\pi}{b}|z_{2}|^{2}<1\Bigr{\\}},$

and by $\varepsilon_{a,b}\in\mathcal{F}(S^{3},\xi_{\mathrm{st}})$ the contact
form which is obtained by pulling the restriction of $\lambda_{0}$ to
$\partial E(a,b)$ back to $S^{3}$ by the radial projection, i.e.

$\epsilon_{a,b}(z)=\left(\frac{\pi}{a}|z_{1}|^{2}+\frac{\pi}{b}|z_{2}|^{2}\right)^{-1}\alpha_{0}(z),\qquad\forall
z=(z_{1},z_{2})\in S^{3}.$

The Reeb flow of $\varepsilon_{a,b}$ has the form

$\phi_{\varepsilon_{a,b}}^{t}(z_{1},z_{2})=\bigl{(}e^{\frac{2\pi}{a}ti}z_{1},e^{\frac{2\pi}{b}ti}z_{2}\bigr{)}.$

The circles $\Gamma_{1}\coloneqq\\{z_{2}=0\\}\subset S^{3}$ and
$\Gamma_{2}\coloneqq\\{z_{1}=0\\}\subset S^{3}$ are closed Reeb orbits of
$\varepsilon_{a,b}$ of period $a$ and $b$, respectively. The Reeb flow of
$\varepsilon_{a,b}$ preserves the 2-tori

$\\{|z_{1}|=\cos\beta,\;|z_{2}|=\sin\beta\\}\subset
S^{3},\qquad\forall\beta\in\left(0,{\textstyle\frac{\pi}{2}}\right),$

on which the flow is periodic if and only if the number $\frac{a}{b}$ is
rational.

The first aim of this Appendix is to present a self-contained proof of the
following result.

######  Theorem A.1.

Let $(a_{0},b_{0})$ be a pair of positive numbers. For every neighborhood $U$
of $(a_{0},b_{0})$ in $(0,+\infty)^{2}$ and every neighborhood $\mathcal{U}$
of $\varepsilon_{a_{0},b_{0}}$ in $\mathcal{F}(S^{3},\xi_{\mathrm{st}})$ there
exist $(a,b)\in U$, a contact form $\alpha$ in $\mathcal{U}$ and a sequence
$(\varphi_{j})$ in $\mathrm{Cont}_{0}(S^{3},\xi_{\mathrm{st}})$, each having
support away from $\Gamma_{1}\cup\Gamma_{2}$, such that:

  1. (i)

$(\varphi_{j}^{*}\varepsilon_{a,b})$ converges to $\alpha$ in the
$C^{\infty}$-topology;

  2. (ii)

the only closed Reeb orbits of $\alpha$ are $\Gamma_{1}$ and $\Gamma_{2}$,
with periods $a$ and $b$, respectively;

  3. (iii)

the Reeb flow of $\alpha$ has a dense orbit.

By (iii), the Reeb flow of $\alpha$ is not conjugate to the one of
$\varepsilon_{a,b}$. Together with (i), this implies that the orbit of
$\varepsilon_{a,b}$ by the action of
$\mathrm{Cont}_{0}(S^{3},\xi_{\mathrm{st}})$ is not $C^{\infty}$-closed in
$\mathcal{F}(S^{3},\xi_{\mathrm{st}})$, see Example 2 in the Introduction. The
above result can be deduced from Theorem A in [Kat73], where Katok used the
conjugacy method which he had introduced together with Anosov in [AK70a] to
construct ergodic Hamiltonian flows arbitrarily close to suitably integrable
flows. Therefore, (iii) can be replaced by the stronger condition:

  1. (iii’)

the Reeb flow of $\alpha$ is ergodic with respect to the invariant measure
induced by the volume form $\alpha\wedge d\alpha$.

Specializing the argument to the perturbation of $\varepsilon_{a_{0},b_{0}}$
and showing just the existence of a dense orbit, instead of ergodicity, allows
us to give here a short self-contained proof, still based on the Anosov-Katok
conjugacy method. See also [CS16] for a reformulation of Katok’s result in the
Reeb setting.

######  Remark A.2.

In the proof of Theorem A.1, the pair $(a,b)$ is obtained as a limit of pairs
of positive numbers $(a_{j},b_{j})$ with rational ratio which is required to
converge extremely fast and we do not control which kind of limiting pairs
$(a,b)$ we can produce. It is actually possible to strengthen the above result
by prescribing the pair $(a,b)$: For any pair of positive numbers $(a,b)$
whose ratio is irrational and not Diophantine, there exist a contact form
$\alpha\in\mathcal{F}(S^{3},\xi_{\mathrm{st}})$ and a sequence $(\varphi_{j})$
in $\mathrm{Cont}_{0}(S^{3},\xi_{\mathrm{st}})$ which satisfy (i), (ii) and
(iii) (or even (iii’)) with respect to $(a,b)$. This follows from a
corresponding result for area-preserving diffeomorphisms of the 2-disk which
is proven in [FS05] and by a construction from [AGZ22], which allows to lift
such diffeomorphisms to Reeb flows on $(S^{3},\xi_{\mathrm{st}})$.

The condition on $(a,b)$ is optimal. Indeed, if the ratio $\frac{a}{b}$ is
rational, then any contact form $\alpha$ which satisfies (i) (even just with
$C^{0}$-convergence) is smoothly conjugate to $\varepsilon_{a,b}$, as
discussed in Example 1 in the Introduction, so cannot satisfy (iii). If the
ratio $\frac{a}{b}$ is irrrational and Diophantine and the contact form
$\alpha$ satisfies (i), then the first-return map to a local Poincaré section
transverse to $\Gamma_{1}$ of the Reeb flow of $\alpha$ is an area-preserving
diffeomorphism with an elliptic and Diophantine fixed point. In his “last
geometric theorem”, Herman proved that arbitrarily near to such a fixed point
there are smooth invariant circles, see [FK09, Theorem 4], and hence the Reeb
flow of $\alpha$ has invariant tori. Since these tori disconnect $S^{3}$,
$\alpha$ cannot satisfy (iii).

Writing the contact form $\alpha$ of Theorem A.1 as $\alpha=f^{2}\alpha_{0}$
for a suitable smooth positive function on $S^{3}$, we consider the starshaped
domain $A\subset\mathbb{C}^{2}$ which is given by

$A\coloneqq\\{rz\mid z\in S^{3},\;0\leq r<f(z)\\}.$

The pullback by the radial projection $S^{3}\rightarrow\partial A$ of
$\lambda_{0}|_{\partial A}$ to $S^{3}$ is then $\alpha$. We can complement
Theorem A.1 with the following result.

######  Theorem A.3.

The domains $A$ and $E(a,b)$ are symplectomorphic but the contact forms which
are given by the restrictions of $\lambda_{0}$ to their boundaries are not
conjugate.

The first examples of pairs of starshaped domains in $\mathbb{C}^{n}$, $n>1$,
having the above properties were constructed by Eliashberg and Hofer in
[EH96]. The construction we present here is somehow simpler, but builds on
some results which are known to hold only in dimension four.

### Proof of theorem A.1.

By slightly perturbing $(a_{0},b_{0})$, we may assume that the ratio
$\frac{a_{0}}{b_{0}}$ is rational. Starting from the pair $(a_{0},b_{0})$ and
from $\varphi_{0}=\mathrm{id}$, we shall inductively construct sequences
$(a_{j},b_{j})$ in $U$ with $\frac{a_{j}}{b_{j}}\in\mathbb{Q}$ and
$(\varphi_{j})$ in $\mathrm{Cont}_{0}(S^{3},\xi_{\mathrm{st}})$ such that
$(a_{j},b_{j})$ converges to some $(a,b)$ in $U$ and the sequence of contact
forms

$\alpha_{j}\coloneqq\varphi_{j}^{*}\varepsilon_{a_{j},b_{j}}$

converges to some contact form $\alpha$ in $\mathcal{U}$. Assuming that
$(a_{j},b_{j})$ and $\varphi_{j}$ have been chosen, we choose $\varphi_{j+1}$
of the form

$\varphi_{j+1}\coloneqq\psi_{j}\circ\varphi_{j},$

for some $\psi_{j}$ in $\mathrm{Cont}_{0}(S^{3},\xi_{\mathrm{st}})$ satisfying
$\psi_{j}^{*}\varepsilon_{a_{j},b_{j}}=\varepsilon_{a_{j},b_{j}}$ whose
properties are discussed below. With these choices, we have for every
$(a^{\prime},b^{\prime})\in(0,+\infty)^{2}$

$\begin{split}&\varphi_{j+1}^{*}\varepsilon_{a^{\prime},b^{\prime}}-\varphi_{j}^{*}\varepsilon_{a_{j},b_{j}}=\varphi_{j+1}^{*}\varepsilon_{a^{\prime},b^{\prime}}-(\psi_{j}^{-1}\circ\varphi_{j+1})^{*}\varepsilon_{a_{j},b_{j}}\\\
&=\varphi_{j+1}^{*}\varepsilon_{a^{\prime},b^{\prime}}-\varphi_{j+1}^{*}((\psi_{j}^{-1})^{*}\varepsilon_{a_{j},b_{j}})=\varphi_{j+1}^{*}(\varepsilon_{a^{\prime},b^{\prime}}-\varepsilon_{a_{j},b_{j}}).\end{split}$
(A.1)

Once $\psi_{j}$, and hence $\varphi_{j+1}$, has be chosen, the above identity
shows that by choosing $(a_{j+1},b_{j+1})$ close to $(a_{j},b_{j})$ we can
make the $C^{k}$-norm of

$\alpha_{j+1}-\alpha_{j}=\varphi_{j+1}^{*}(\varepsilon_{a_{j+1},b_{j+1}}-\varepsilon_{a_{j},b_{j}})$

arbitrarily small. Therefore, we can ensure the convergence of $(a_{j},b_{j})$
to some $(a,b)\in U$ and of $(\alpha_{j})$ to some $\alpha\in\mathcal{U}$ in
the $C^{\infty}$-topology. Moreover, (A.1) shows that by making the
convergence of $(a_{j},b_{j})$ fast enough and by a diagonal argument we can
also ensure that the sequence $(\varphi_{j}^{*}\varepsilon_{a,b})$ converges
to $\alpha$, as claimed in (i).

Next, we remark that any open condition which is satisfied by $\alpha_{j}$ can
be transmitted to $\alpha$ by making the convergence of $(a_{j},b_{j})$
sufficiently fast. More precisely: if $\mathcal{V}_{j}$ is some open
neighborhood of $\alpha_{j}$ in $\mathcal{F}(S^{3},\xi_{\mathrm{st}})$, we can
fix a closed neighborhood $\mathcal{W}_{j}\subset\mathcal{V}_{j}$ of
$\alpha_{j}$ and make all the subsequent choices of $(a_{i},b_{i})$ in such a
way that for every $i>j$ the contact form $\alpha_{i}$ belongs to
$\mathcal{W}_{j}$. By doing this, we can ensure that also the limiting contact
form $\alpha$ belongs to $\mathcal{V}_{j}$.

We now discuss the choice of $\psi_{j}$. By choosing each $\psi_{j}$ to be
supported away from the link $\Gamma\coloneqq\Gamma_{1}\cup\Gamma_{2}$, we can
ensure that $\alpha$ agrees up to every order with $\varepsilon_{a,b}$ on
$\Gamma$, and hence $\Gamma_{1}$ and $\Gamma_{2}$ are closed Reeb orbits of
$\alpha$ of period $a$ and $b$, respectively.

Write $\frac{a_{j}}{b_{j}}=\frac{p_{j}}{q_{j}}$, where $p_{j}$ and $q_{j}$ are
coprime natural numbers. By choosing $(a_{j+1},b_{j+1})$ close to
$(a_{j},b_{j})$, but not equal to it, we have that the sequences $(p_{j})$ and
$q_{j})$ diverge.

We claim that if the convergence of $(a_{j},b_{j})$ is sufficiently fast then
$\Gamma_{1}$ and $\Gamma_{2}$ are the only closed Reeb orbits of $\alpha$. Let
$V_{j}$ be a sequence of open neighborhoods of $\Gamma$ whose intersection is
$\Gamma$. Since the Reeb orbits of $\alpha_{j}$ other than $\Gamma_{1}$ and
$\Gamma_{2}$ are all closed with period $p_{j}b_{j}=q_{j}a_{j}$, the Reeb flow
of $\alpha_{j}$ has no closed orbits of period $\tau\leq\tau_{j}\coloneqq
p_{j}b_{j}-1$ which pass through $S^{3}\setminus V_{j}$. The set of contact
forms with the latter property is $C^{1}$-open, so by the remark above we can
guarantee that the Reeb flow of $\alpha$ has no closed orbits of period
$\tau\leq\tau_{j}$ which pass through $S^{3}\setminus V_{j}$, for every
$j\in\mathbb{N}$. Since the sequence $(V_{j})$ shrinks to $\Gamma$ and
$(\tau_{j})$ diverges, $\Gamma_{1}$ and $\Gamma_{2}$ are the only two closed
Reeb orbits of $\alpha$, as claimed in (ii).

Fix some infinitesimal sequence $(\epsilon_{j})$ of positive numbers. We claim
that by a suitable choice of the defining data we can ensure that for every
$j\geq 1$ the Reeb flow of $\alpha_{j}$ has an orbit which is
$\epsilon_{j}$-dense. Here, a subset of $S^{3}$ is said to be $\epsilon$-dense
if it meets any open ball of radius $\epsilon$, with respect to some fixed
metric $d$ on $S^{3}$.

Indeed, once this is achieved we can argue as follows. Having an
$\epsilon_{j}$-dense orbit is a $C^{0}$-open condition in the space of smooth
vector fields, and hence a $C^{1}$-open condition in
$\mathcal{F}(S^{3},\xi_{\mathrm{st}})$. Therefore, the above property of each
$\alpha_{j}$ can be transmitted to $\alpha$, whose Reeb flow then has an
$\epsilon_{j}$-dense orbit for every $j$. When non-empty, the set $O_{j}$ of
$\epsilon_{j}$-dense orbits of a flow is open and $\epsilon_{j}$-dense. Using
that $\epsilon_{j}\to 0$, a standard argument, which is analogous to the proof
of Baire’s theorem, shows that $\cap_{k\in\mathbb{N}}O_{j_{k}}$ is non-empty
for a suitable subsequence $(j_{k})$. Every element of this intersection has a
dense orbit and, therefore, $\alpha$ satisfies (iii).

There remains to prove the following claim, in which $\epsilon$ is an
arbitrary positive number: Once $\varphi_{j}$ and $(a_{j},b_{j})$ have been
chosen, by a suitable choice of the diffeomorphism $\psi_{j}$, which we recall
is required to preserve the contact form $\varepsilon_{a_{j},b_{j}}$ and to be
supported away from $\Gamma$, and by choosing $(a_{j+1},b_{j+1})$ sufficiently
close to $(a_{j},b_{j})$, we can make sure that the Reeb flow of

$\alpha_{j+1}=\varphi_{j}^{*}(\psi_{j}^{*}\varepsilon_{a_{j+1},b_{j+1}})$

has an orbit which is $\epsilon$-dense. The proof of this fact uses the
following lemma.

######  Lemma A.4.

Let $(a,b)$ be a pair of positive numbers with rational ratio. Let
$(\gamma_{1},\dots,\gamma_{k})$ and
$(\gamma_{1}^{\prime},\dots,\gamma_{k}^{\prime})$ be $k$-tuples of distinct
Reeb orbits of $\varepsilon_{a,b}$ in $S^{3}\setminus\Gamma$. Then there
exists $\psi\in\mathrm{Cont}_{0}(S^{3},\xi_{\mathrm{st}})$ which preserves the
contact form $\varepsilon_{a,b}$, is supported away from $\Gamma$ and
satisfies $\psi(\gamma_{j})=\gamma_{j}^{\prime}$ for every $j=1,\dots,k$.

Postponing the proof of this lemma, we show how it implies the above claim.
Note that the Reeb flow of $\alpha_{j+1}$ has the form

$\phi_{\alpha_{j+1}}^{t}=\varphi_{j}^{-1}\circ\psi_{j}^{-1}\circ\phi^{t}_{\varepsilon_{a_{j+1},b_{j+1}}}\circ\psi_{j}\circ\varphi_{j},$
(A.2)

where $\phi^{t}_{\varepsilon_{a_{j+1},b_{j+1}}}$ denotes the Reeb flow of
$\varepsilon_{a_{j+1},b_{j+1}}$. Let $\delta>0$ be such that

$d(z,z^{\prime})<\delta\qquad\Rightarrow\qquad
d\bigl{(}\varphi_{j}^{-1}(z),\varphi_{j}^{-1}(z^{\prime}))<\epsilon,$ (A.3)

and let $A\subset S^{3}\setminus\Gamma$ be a finite set which is
$\frac{\delta}{2}$-dense in $S^{3}$. Fix some invariant torus $T$ in
$S^{3}\setminus\Gamma$, such as for instance

$T\coloneqq\bigl{\\{}(z_{1},z_{2})\in
S^{3}\mid|z_{1}|^{2}=|z_{2}|^{2}=\textstyle{\frac{1}{2}}\bigr{\\}}.$

Since this torus consists of infinitely many Reeb orbits of
$\varepsilon_{a_{j},b_{j}}$, by the above lemma there exists
$\psi_{j}\in\mathrm{Cont}_{0}(S^{3},\xi_{\mathrm{st}})$ which preserves
$\varepsilon_{a_{j},b_{j}}$ and is supported away from $\Gamma$ such that
$\psi_{j}^{-1}(T)$ contains the finite set $A$, and is therefore
$\frac{\delta}{2}$-dense in $S^{3}$.

Now let $\sigma>0$ be such that

$d(z,z^{\prime})<\sigma\qquad\Rightarrow\qquad
d\bigl{(}\psi_{j}^{-1}(z),\psi_{j}^{-1}(z^{\prime}))<\textstyle{\frac{\delta}{2}}.$
(A.4)

Let $(a^{\prime},b^{\prime})$ be a pair of positive numbers with rational
ratio $\frac{a^{\prime}}{b^{\prime}}=\frac{p}{q}$, where $p$ and $q$ are
coprime natural numbers. Each Reeb orbit of
$\varepsilon_{a^{\prime},b^{\prime}}$ in $T$ is $\sigma$-dense, provided that
$p$ and $q$ are large enough. Therefore, by choosing $(a_{j+1},b_{j+1})$ with
the already required properties and sufficiently close to $(a_{j},b_{j})$, we
obtain that each Reeb orbit of $\varepsilon_{a_{j+1},b_{j+1}}$ is
$\sigma$-dense in $T$.

Let $z$ be a point in $\varphi_{j}^{-1}\circ\psi_{j}^{-1}(T)$ and set
$w\coloneqq\psi_{j}\circ\varphi_{j}(z)\in T$. Then the orbit of $w$ by the
Reeb flow of $\varepsilon_{a_{j+1},b_{j+1}}$ is $\sigma$-dense in $T$. By
(A.4), the image of this orbit by $\psi_{j}^{-1}$ is $\frac{\delta}{2}$-dense
in $\psi_{j}^{-1}(T)$ and hence $\delta$-dense in $S^{3}$. Together with (A.2)
and (A.3), we conclude that the orbit of $z$ by the Reeb flow of
$\alpha_{j+1}$ is $\epsilon$-dense. This concludes the proof of Theorem A.1.

###### Proof of Lemma A.4.

Since $\frac{a}{b}$ is rational, the Reeb flow of $\varepsilon_{a,b}$ defines
a free $S^{1}$-action on $S^{3}\setminus\Gamma$. The orbit space of this
action is a smooth surface $\Sigma$ which is endowed with a symplectic form
$\omega$ satisfying $\pi^{*}\omega=d\varepsilon_{a,b}$, where

$\pi:S^{3}\setminus\Gamma\rightarrow\Sigma$

denotes the quotient projection. Let $(p_{1},\dots,p_{k})$ and
$(p_{1}^{\prime},\dots,p_{k}^{\prime})$ be the $k$-tuples of distinct points
in $\Sigma$ such that

$\gamma_{j}=\pi^{-1}(p_{j}),\qquad\gamma_{j}^{\prime}=\pi^{-1}(p_{j}^{\prime})\qquad\forall
j=1,\dots,k.$

It is easy to show that the group of compactly supported Hamiltonian
diffeomorphisms of $(\Sigma,\omega)$ acts $k$-transitively: There exists a
smooth function $H:[0,1]\times\Sigma\rightarrow\mathbb{R}$ with compact
support such that the time-1 map of the induced Hamiltonian flow maps $p_{j}$
to $p_{j}^{\prime}$, for every $j=1,\dots,k$ (see e.g. [Ban97, p. 109]). Lift
$H$ to a smooth function $K:[0,1]\times S^{3}\rightarrow\mathbb{R}$ supported
away from $\Gamma$ and let $X$ be the contact vector field on $S^{3}$ which is
induced by the Hamiltonian $K$ and the contact form $\varepsilon_{a,b}$, i.e.,
the unique vector field satisfying

$\imath_{X}d\varepsilon_{a,b}=\bigl{(}\imath_{R_{\varepsilon_{a,b}}}dK)\varepsilon_{a,b}-dK,\qquad\imath_{X}\varepsilon_{a,b}=K,$

where $R_{\varepsilon_{a,b}}$ denote the Reeb vector field of
$\varepsilon_{a,b}$. The flow of $X$ consists of contactomorphisms of
$(S^{3},\xi_{\mathrm{st}})$ preserving $\varepsilon_{a,b}$ which are supported
away from $\Gamma$ and whose time-1 map $\psi$ maps $\gamma_{j}$ to
$\gamma_{j}^{\prime}$, for every $j=1,\dots,k$. ∎

### Proof of theorem A.3.

By composition with the radial projections, a diffeomorphism from $\partial
E(a,b)$ to $\partial A$ which intertwines the restrictions of $\lambda_{0}$
would induce a smooth conjugacy between $\varepsilon_{a,b}$ and $\alpha$ on
$S^{3}$, which cannot exist because the Reeb flow of $\alpha$ has a dense
orbit while the one of $\varepsilon_{a,b}$ does not. So we only have to prove
that $E(a,b)$ and $A$ are symplectomorphic.

Set for simplicity $E\coloneqq E(a,b)$. Write
$\varphi_{j}^{*}\varepsilon_{a,b}=f_{j}^{2}\,\alpha_{0}$ and set

$A_{j}\coloneqq\\{rz\mid z\in S^{3},\;0\leq r<f_{j}(z)\\}.$

By composing the contactomorphism $\varphi_{j}^{-1}:S^{3}\rightarrow S^{3}$,
which isisotopic to the identity, with the radial projections $\partial
E\rightarrow S^{3}$ and $S^{3}\rightarrow\partial A_{j}$, we obtain a
diffeomorphism from $\partial E$ to $\partial A_{j}$ which intertwines the
restrictions of $\lambda_{0}$. The positively 1-homogeneous extension of this
map is a symplectomorphism of $\mathbb{C}^{2}\setminus\\{0\\}$ isotopic to the
identity via such maps, and mapping $rE\setminus\\{0\\}$ to
$rA_{j}\setminus\\{0\\}$, for every $r>0$. Repeating the argument at the end
of the proof of Theorem 1 (ii) in Section 6, this map can be smoothed near the
origin producing a global symplectomorphism
$\phi_{j}:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ such that

$\phi_{j}(rE)=rA_{j}\qquad\forall r\geq\textstyle{\frac{1}{2}}.$

The $C^{0}$-convergence of $(f_{j})$ to $f$ implies that, up to passing to a
subsequence, we can find a strictly increasing sequence $(r_{j})$ in the
interval $(\frac{1}{2},1)$ which converges to 1 and satisfies

$r_{j}\overline{A}_{j}\subset r_{j+1}A_{j+1},\qquad\bigcup_{j\geq
0}r_{j}A_{j}=A.$ (A.5)

Starting with $\psi_{0}\coloneqq\phi_{0}$ and arguing inductively, we wish to
construct symplectomorphisms
$\psi_{j}:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ such that for every $j\geq
0$:

$\displaystyle\psi_{j+1}=\psi_{j}\quad$ $\displaystyle\mbox{on }r_{j}E,$ (A.6)
$\displaystyle\psi_{j+1}=\phi_{j+1}\quad$ $\displaystyle\mbox{on
}\mathbb{C}^{2}\setminus r_{j+1}E.$ (A.7)

Once $\psi_{0},\dots,\psi_{j}$ have been constructed, the construction of
$\psi_{j+1}$ goes as follows. The symplectomorphism
$\theta\coloneqq\phi_{j+1}^{-1}\circ\psi_{j}$ maps $r_{j}E$ to the domain
$\phi_{j+1}^{-1}(r_{j}A_{j})$, whose closure satisfies

$\overline{\phi_{j+1}^{-1}(r_{j}A_{j})}=\phi_{j+1}^{-1}(r_{j}\overline{A_{j}})\subset\phi_{j+1}^{-1}(r_{j+1}A_{j+1})=r_{j+1}E.$

Since the space of symplectic embeddings between ellipsoids is connected, see
[McD09], we can find a symplectic isotopy
$\theta_{t}:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ such that
$\theta_{0}=\mathrm{id}$, $\theta_{1}=\theta$ and

$\theta_{t}(r_{j}\overline{E})\subset r_{j+1}E\qquad\forall t\in[0,1].$

Let $H_{t}:\mathbb{C}^{2}\rightarrow\mathbb{R}$ be a smooth path of
Hamiltonians generating $\theta_{t}$. Let
$\chi:\mathbb{C}^{2}\rightarrow\mathbb{R}$ be a smooth function with support
in $r_{j+1}E$ and taking the value 1 on
$\bigcup_{t\in[0,1]}\theta_{t}(r_{j}\overline{E})$. Denote by
$\eta:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ the time-1 map of the
Hamiltonian isotopy induced by the non-autonomous Hamiltonian $\chi H$. Then

$\eta=\theta_{1}=\phi_{j+1}^{-1}\circ\psi_{j}\quad\mbox{on
}r_{j}E,\qquad\eta=\mathrm{id}\quad\mbox{on }\mathbb{C}^{2}\setminus
r_{j+1}E,$

and hence $\psi_{j+1}\coloneqq\phi_{j+1}\circ\eta$ satisfies (A.6) and (A.7).

Note that by (A.7) we have

$\psi_{j}(rE)=rA_{j}\quad\forall r\geq r_{j},\qquad\forall j\geq 0.$ (A.8)

By (A.6), the sequence $\psi_{j}(z)$ stabilizes for every $z\in E$, and hence
the map

$\psi:E\rightarrow\mathbb{C}^{2},\qquad\psi(z)\coloneqq\lim_{j\rightarrow\infty}\psi_{j}(z)$

is a well defined symplectic embedding, which by (A.5) and (A.8) has image
$A$. The maps $\psi$ is then the required symplectomorphism from $E(a,b)$ to
$A$.

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