# Equidistributed Closed Geodesics on Closed Finsler and Riemannian Surfaces

Hui Liu<EMAIL_ADDRESS>School of Mathematics and Statistics, Wuhan
University, China  and  Lei Liu<EMAIL_ADDRESS>Beijing International
Center for Mathematical Research, Peking University, China

###### Abstract.

In this paper, we prove that for a $C^{\infty}$-generic Finsler (symmetric
Finsler and Riemannian) metric on a closed surface $\Sigma$, there exists a
sequence of (nondegenerate) closed geodesics which are equidistributed on
$\Sigma$. The proof is based on the volume property of the spectral invariants
in embedded contact homology theory by Cristofaro-Gardiner, Hutchings, Ramos
and some transversality arguments, which are inspired by the equidistribution
result of Irie for three-dimensional Reeb flows and the analogous result of
Marques, Neves and Song for embedded minimal hypersurfaces.

The first author was supported by NSFC(Nos. 12022111,11771341).

AMS Subject Classification: 70H12, 53D25, 53D42.

Key Words: Reeb flow, Finsler and Riemanian surface, closed geodesics,
equidistribution, embedded contact homology.

##  1\. Introduction

###  1.1. Finsler and Riemannian settings

Let $(\Sigma,F)$ be a Finsler manifold, where $F$ is the Finsler metric
satisfying

  * (I)

$F:T\Sigma\setminus\Sigma\rightarrow\mathbb{R}_{+}$ is a smooth function and
$F|_{\Sigma}=0$.

  * (II)

$F(x,av)=aF(x,v)$ for any $(x,v)\in T\Sigma\setminus\Sigma$ and $a>0$.

  * (III)

$(v_{1},v_{2})\mapsto\frac{d^{2}}{dsdt}|_{s=t=0}F^{2}(x,v+sv_{1}+tv_{2})$ is
positive definite.

We call $F$ symmetric if

  * (II’)

$F(x,av)=|a|F(x,v)$ for any $(x,v)\in T\Sigma\setminus\Sigma$ and $a\neq 0$.

Denote by $\mathcal{F}(\Sigma)$ (resp. $\mathcal{F}^{s}(\Sigma)$) the space of
all Finsler (resp. symmetric Finsler) metrics on $\Sigma$. By fixing a Finsler
metric $F$, we can also rewrite

$\mathcal{F}(\Sigma)=\\{fF\in\mathcal{F}(\Sigma)|\
f=F^{-1}(\cdot)F^{\prime}(\cdot)\in
C^{\infty}(\mathbb{S}\Sigma,\mathbb{R}_{+}),\ \forall
F^{\prime}\in\mathcal{F}(\Sigma)\\},$

where $\mathbb{S}\Sigma$ is the sphere bundle over $\Sigma$ and
$f(av)=\frac{F^{\prime}(av)}{F(av)}=\frac{F^{\prime}(v)}{F(v)}=f(v)$ for any
$a>0$. $\mathcal{F}^{s}(\Sigma)$ can be rewritten in a similar way. The
$C^{\infty}$-topologies of $\mathcal{F}(\Sigma)$ and $\mathcal{F}^{s}(\Sigma)$
are induced by

(1.1)
$\|f_{1}-f_{2}\|_{C^{\infty}}:=\sum^{\infty}_{k=0}2^{-k}\frac{\|\nabla^{k}(f_{1}-f_{2})\|_{C^{0}}}{1+\|\nabla^{k}(f_{1}-f_{2})\|_{C^{0}}},\
\forall f_{1},f_{2}\in C^{\infty}(\mathbb{S}\Sigma,\mathbb{R}).$

Since $F^{2}$ is strictly convex on $T\Sigma$, by using the Legendrian
transformation, there is a one to one correspondence

(1.2) $\mathcal{L}_{F}:T\Sigma\rightarrow
T^{*}\Sigma\left(v=v_{1}\frac{\partial}{\partial
q_{1}}+v_{2}\frac{\partial}{\partial q_{2}}\mapsto F(v)\ (\frac{\partial
F}{\partial v_{1}}(v)dq_{1}+\frac{\partial F}{\partial
v_{2}}(v)dq_{2})\right),$

which is independent of the choice of local coordinates $(q_{1},q_{2})$. Since
$\mathcal{L}_{F}$ satisfies $\mathcal{L}_{F}(av)=a\mathcal{L}_{F}(v)$ for any
$a>0$ (or for any $a\neq 0$ if $F$ is symmetric, see (2.3)), we can introduce
a dual metric $F^{*}(p)=F(\mathcal{L}_{F}^{-1}p)$ on $T^{*}\Sigma$ with
satisfying (I) to (III) and (II’), similarly. Denote by
$S^{*}_{F}\Sigma=\\{(x,p)\in T^{*}\Sigma|\ F^{*}(p)=1\\}$ the unit cotangent
bundle and let $\lambda(F)_{e}=e\circ\pi_{*},\ \forall e\in S^{*}_{F}\Sigma$,
be the canonical Liouville form, where $\pi_{*}$ is the differential of
projection $\pi:T^{*}\Sigma\rightarrow\Sigma$. Then by Subsection 2.2,
$(S^{*}_{F}\Sigma,\xi_{F}=\ker\lambda(F))$ forms a contact manifold and
satisfies
$(S^{*}_{F}\Sigma,{F^{\prime}}^{*}(\cdot)^{-1}\lambda(F))\cong(S^{*}_{F^{\prime}}\Sigma,\lambda(F^{\prime}))$
for any $F,F^{\prime}\in\mathcal{F}(\Sigma)$. Now we define

(1.3) $\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F}):=\\{\mathrm{contact\ forms}\
\lambda_{F^{\prime}}={F^{\prime}}^{*}(\cdot)^{-1}\lambda(F),\ \forall
F^{\prime}\in\mathcal{F}(\Sigma)\\},$

and similarly define $\Lambda^{sFin}(S^{*}_{F}\Sigma,\xi_{F})$ for any
$F^{\prime}\in\mathcal{F}^{s}(\Sigma)$ by choosing
$F\in\mathcal{F}^{s}(\Sigma)$. Its $C^{\infty}$-topology is induced by

$\|\lambda_{F_{1}}-\lambda_{F_{2}}\|:=\|F^{-1}F_{1}-F^{-1}F_{2}\|_{C^{\infty}(\mathbb{S}\Sigma,\mathbb{R})},\
\forall F_{1},F_{2}\in\mathcal{F}(\Sigma).$

Then $\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$ (resp.
$\Lambda^{sFin}(S^{*}_{F}\Sigma,\xi_{F})$) and $\mathcal{F}(\Sigma)$ (resp.
$\mathcal{F}^{s}(\Sigma)$) are homeomorphic to each other.

As a special case, we denote by
$\mathcal{M}(\Sigma)\subset\mathcal{F}^{s}(\Sigma)(g(\cdot,\cdot)\hookrightarrow
F_{g}:=\|\cdot\|_{g})$ the space of Riemannian metrics on $\Sigma$. Then by
fixing $g(\cdot,\cdot)\in\mathcal{M}(\Sigma)$, one can rewrite

$\mathcal{M}(\Sigma)=\\{(f_{g^{\prime}})^{2}g\in\mathcal{M}(\Sigma)|\
f_{g^{\prime}}(\cdot)=\|\cdot\|_{g}^{-1}\|\cdot\|_{g^{\prime}}\in
C^{\infty}(\mathbb{S}\Sigma,\mathbb{R}_{+}),\ \forall
g^{\prime}\in\mathcal{M}(\Sigma)\\}.$

The $C^{\infty}$-topology of $\mathcal{M}(\Sigma)$ is also induced by (1.1).
By definition (1.2), we can conclude that
$\mathcal{L}_{F_{g}}(v)=(v,\cdot)_{g}$, then we rewrite
$\mathcal{L}_{g}(\cdot):=\mathcal{L}_{F_{g}}(\cdot)$ and define the dual
metric $g^{*}$ on $T^{*}\Sigma$ as $\|\cdot\|_{g^{*}}:=F_{g}^{*}(\cdot)$,
which satisfies $\|\mathcal{L}_{g}(v)\|_{g^{*}}=\|v\|_{g}$. Then we obtain a
contact manifold
$(S^{*}_{g}\Sigma,\xi_{g}=\ker\lambda(g)):=(S^{*}_{F_{g}}\Sigma,\xi_{F_{g}})$,
where $\lambda(g)$ is the Liouville form $\lambda(F_{g})$ on $g^{*}$-unit
cotangent bundle $S^{*}_{g}\Sigma$. Similar to
$\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$, we define

(1.4) $\Lambda^{Rie}(S^{*}_{g}\Sigma,\xi_{g}):=\\{\mathrm{contact\ forms}\
\lambda_{g^{\prime}}=\|\cdot\|^{-1}_{{g^{\prime}}^{*}}\lambda(g),\ \forall
g^{\prime}\in\mathcal{M}(\Sigma)\\}.$

Its $C^{\infty}$-topology is induced by

$\|\lambda_{g_{1}}-\lambda_{g_{2}}\|:=\|f_{g_{1}}-f_{g_{2}}\|_{C^{\infty}(\mathbb{S}\Sigma,\mathbb{R})},\
\forall g_{1},g_{2}\in\mathcal{M}(\Sigma).$

Then $\Lambda^{Rie}(S^{*}_{g}\Sigma,\xi_{g})$ and $\mathcal{M}(\Sigma)$ are
homeomorphic to each other, either.

Now we denote by $\mathrm{CG}(\Sigma,F)$ (resp. $\mathrm{CG}(\Sigma,g)$) the
collection of all closed geodesics on $(\Sigma,F)$ (resp. $(\Sigma,g)$). A
Finsler (resp. Riemannian) 1-current on $(\Sigma,F)$ (resp. $(\Sigma,g)$) is a
formal linear combination $a_{1}\sigma_{1}+\cdots+a_{k}\sigma_{k}$, with
$a_{i}>0$ and $\sigma_{i}\in\mathrm{CG}(\Sigma,F)$ (resp.
$\mathrm{CG}(\Sigma,g)$) for any $i=1,\cdots,k$. Denote by
$l_{\mathrm{CG}}(\Sigma,F)$ (resp. $l_{\mathrm{CG}}(\Sigma,g)$) the collection
of all Finsler (resp. Riemannian) 1-currents. Similarly, let $R(Y,\lambda)$ be
the collection of all Reeb orbits with respect to $\lambda$ (see Section 2.1
for definition of Reeb orbits) and let $L(Y,\lambda)$ (resp.
$L_{\mathbb{Z}_{+}}(Y,\lambda)$) be the collection of Reeb 1-currents with
$\mathbb{R}_{+}$ (resp. $\mathbb{Z}_{+}$)-coefficient, i.e. finite
$\mathbb{R}_{+}$ (resp. $\mathbb{Z}_{+}$)-linear combination of elements in
$R(Y,\lambda)$. Then the projection
$R(S^{*}_{F}\Sigma,\lambda(F))\rightarrow\mathrm{CG}(\Sigma,F)(\gamma\mapsto\pi(\gamma))$
is a one to one correspondence. So is the projection from
$L(S^{*}_{F}\Sigma,\lambda(F))$ to $l_{\mathrm{CG}}(\Sigma,F)$. We denote by
$\tilde{\sigma}=(\sigma,\dot{\sigma})$ and
$\gamma_{\sigma}=\mathcal{L}_{F}(\tilde{\sigma})$ the corresponding curves in
$S_{F}\Sigma$ and $S^{*}_{F}\Sigma$, respectively, for any
$\sigma\in\mathrm{CG}(\Sigma,F)$. Moreover, if $\lambda=\lambda_{F^{\prime}}$
belongs to $\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$ for some
$F^{\prime}\in\mathcal{F}(\Sigma)$, then $L(S^{*}_{F}\Sigma,\lambda)$ is one
to one corresponding to $l_{\mathrm{CG}}(\Sigma,F^{\prime})$. Since
$\mathcal{M}(\Sigma)\subset\mathcal{F}^{s}(\Sigma)\subset\mathcal{F}(\Sigma)$,
then
$\Lambda^{Rie}(S^{*}_{g}\Sigma,\xi_{g})\subset\Lambda^{sFin}(S^{*}_{F_{g}}\Sigma,\xi_{F_{g}})\subset\Lambda^{Fin}(S^{*}_{F_{g}}\Sigma,\xi_{F_{g}})$,
and we also have one to one correspondence
$R(S^{*}_{g}\Sigma,\lambda(g))\rightarrow\mathrm{CG}(\Sigma,g)(\gamma_{\sigma}\mapsto\sigma)$
and so is the projection from $L(S^{*}_{g}\Sigma,\lambda(g))$ to
$l_{\mathrm{CG}}(\Sigma,g)$.

From another perspective, the Finsler and Reeb 1-currents are also elements in
dual space of $\Omega^{1}(S_{F}\Sigma)$ and $\Omega^{1}(S^{*}_{F}\Sigma)$,
respectively, i.e.

(1.5)
$\sum_{i=1}^{k}a_{i}\sigma_{i}(\alpha)=\sum_{i=1}^{k}a_{i}\int_{\tilde{\sigma}_{i}}\alpha,\
\sum_{i=1}^{k}a_{i}\gamma_{i}(\beta)=\sum_{i=1}^{k}a_{i}\int_{\gamma_{i}}\beta,$

for any $\alpha\in\Omega^{1}(S_{F}\Sigma)$ and
$\beta\in\Omega^{1}(S^{*}_{F}\Sigma)$.

###  1.2. History and Main results

In this paper, we are going to study the distribution problem of closed
geodesics on closed Finsler and Riemannian surfaces. This problem can be
traced back to the work of Hadamard [14] at the end of the 19th century.
Combining the Poincaré recurrence theorem, he showed that the closed geodesics
are densely distributed in any negatively curved closed Riemannian manifolds.
This can also be proved by Anosov Closing Lemma with the fact that the
geodesic flow on negatively curved closed Riemannian manifold is an “Anosov”
flow, (see [1] or Theorem 6.4.15 and 17.6.2 in [19]). For more general
denseness or equidistribution results of closed geodesics, we refer to [3, 4,
8, 28].

For general cases, the distribution problem of closed geodesics have been
studied for both Riemannian and Finsler cases. In 2015, by using volume
property of spectral invariants of ECH(embedded contact homology) established
by Cristofaro-Gardiner, Hutchings, Ramos in [9], Irie [16] proved the
denseness of Reeb orbits for generic contact forms on any three dimensional
contact manifold and also the denseness of closed geodesics for generic
Riemannian metrics on any closed surface. Later on, inspired by Marques,
Neves, Song [23] about the distribution problem of embedded minimal
hypersurfaces, Irie [17] found an equidistributed sequence of Reeb orbits for
generic contact forms on arbitrary closed three dimensional contact manifold.
However, it can not directly implies the distribution result of closed
geodesics on Finsler and Riemannian surfaces, and the Riemannian case is more
subtle. In 2019, Chen [7] applied the local perturbation method for Finsler
metrics constructed in [5] and showed that for any closed surface $\Sigma$,
the closed geodesics are dense for $C^{\infty}$-generic (symmetric) Finsler
metrics. Recently, Li and Staffa [20] proved an equidistribution result of
closed geodesics on any closed Riemannian surface for a generic Riemannian
metric by using the “Weyl law” of Volume spectrum established by Liokumovich,
Marques, Neves [21]. In this paper, we prove some equidistribution results of
closed geodesics on any closed (symmetric) Finsler surface and Riemannian
surface, which gives a quantitative refinement of the $C^{\infty}$-generic
density theorem of Chen [7], and generalize the result of Li and Staffa into
the Finsler setting, while our method are inspired by Irie [17] and completely
different from Li and Staffa [20].

We state the main results as follows.

######  Theorem 1.1.

Let $F$ be any element in $\mathcal{F}^{s}(\Sigma)$. For generic
$\lambda\in\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$ or
$\lambda\in\Lambda^{sFin}(S^{*}_{F}\Sigma,\xi_{F})$, there exists a sequence
of Reeb 1-currents $\\{C_{N}\\}_{N\geq 1}\subset L(S^{*}_{F}\Sigma,\lambda)$
such that

(1.6)
$\lim_{N\rightarrow+\infty}C_{N}(f\lambda)=d\lambda(f\lambda):=\int_{S^{*}_{F}\Sigma}f\lambda\wedge
d\lambda,\ \forall f\in C^{\infty}(\Sigma,\mathbb{R}).$

######  Theorem 1.2.

Let $g$ be any element in $\mathcal{M}(\Sigma)$. For generic
$\lambda\in\Lambda^{Rie}(S^{*}_{g}\Sigma,\xi_{g})$, there exists a sequence of
Reeb 1-currents $\\{C_{N}\\}_{N\geq 1}\subset L(S^{*}_{g}\Sigma,\lambda)$ such
that

(1.7)
$\lim_{N\rightarrow+\infty}C_{N}(f\lambda)=d\lambda(f\lambda):=\int_{S^{*}_{g}\Sigma}f\lambda\wedge
d\lambda,\ \forall f\in C^{\infty}(\Sigma,\mathbb{R}).$

Let $l_{F}(\cdot)$ be the length function on $\mathrm{CG}(\Sigma,F)$. If
$F^{2}=g\in\mathcal{M}(\Sigma)$, we denote by $l_{g}(\cdot)=l_{F}(\cdot)$. As
a conclusion of Theorem 1.1 and 1.2, we have the following equidistribution
results.

######  Theorem 1.3.

For generic $F$ in $\mathcal{F}(\Sigma)$ or $\mathcal{F}^{s}(\Sigma)$, there
exists a sequence of closed geodesics $\\{\sigma_{k}\\}_{k\geq 1}$ on $\Sigma$
such that

(1.8)
$\lim_{k\rightarrow+\infty}\frac{\sigma_{1}(f\beta(F))+\cdots+\sigma_{k}(f\beta(F))}{l_{F}(\sigma_{1})+\cdots+l_{F}(\sigma_{k})}=\frac{\int_{\Sigma}fd\mathrm{vol}_{F}}{\mathrm{vol}(\Sigma,F)},\
\forall f\in C^{\infty}(\Sigma,\mathbb{R}),$

where $\beta(F)$ is the Hilbert form on $S_{F}\Sigma$ in (2.4) and
$d\mathrm{vol}_{F}$ is the Finsler volume form in (2.6).

Note that one cannot expect the equidistribution result of the above theorem
holds for every Finsler metric, since the famous example constructed by A.
Katok [18] on Finsler 2-sphere shows that an irreversible (non-symmetric)
Finsler $S^{2}$ may have exactly two geometrically distinct closed geodesics.
In fact, Bangert and Long [2] proved that there always exist at least two
geometrically distinct closed geodesics on every Finsler $S^{2}$.

######  Theorem 1.4.

For generic $g\in\mathcal{M}(\Sigma)$, there exists a sequence of closed
geodesics $\\{\sigma_{k}\\}_{k\geq 1}$ on $\Sigma$ such that

(1.9)
$\lim_{k\rightarrow+\infty}\frac{\sigma_{1}(fds(g))+\cdots+\sigma_{k}(fds(g))}{l_{g}(\sigma_{1})+\cdots+l_{g}(\sigma_{k})}=\frac{\int_{\Sigma}fd\mathrm{vol}_{g}}{\mathrm{vol}(\Sigma,g)},\
\forall f\in C^{\infty}(\Sigma,\mathbb{R}),$

where $s(g)$ denotes the arc length parameter on any
$\sigma\in\mathrm{CG}(\Sigma,g)$.

######  Remark 1.1.

In Theorem 1.3 (similar to 1.4), the generic metric $F\in\mathcal{F}(\Sigma)$
may not be bumpy and the equidistributed closed geodesics
$\\{\sigma_{i}\\}_{i\geq 1}\subset\mathrm{CG}(\Sigma,F)$ can be all
nondegenerate and arise from certain representations $\\{C_{k}\\}\subset
L_{\mathbb{Z}_{+}}(S^{*}_{F}\Sigma,\lambda(F))$ of ECH spectral invariants
$c_{\varsigma_{k}}(S^{*}_{F}\Sigma,\lambda(F))$ with repeating the closed
geodesics within $C_{k}$ more and more times as $k\rightarrow+\infty$, where
$\\{\varsigma_{k}\\}_{k\geq 1}$ are given by Proposition 2.1 (5), i.e. the
volume property of ECH spectral invariants.

###  1.3. Sketch of the proof

The proof of Theorem 1.2 (resp. Theorem 1.4), i.e. Riemannian case, is similar
to the proof of Theorem 1.1 (resp. Theorem 1.3), i.e. Finsler case, since the
lemmas in Section 3 are available for both Finsler and Riemannian case. Once
we have proved Theorem 1.1, then by using the approximating and selecting
argument in [17] and combining the relationships between the geodesic flow and
Reeb flow, we can prove Theorem 1.3. Thus we only sketch the proof of Theorem
1.1, which adopt some ideas of Irie in [17], but work in the Finsler and
Riemannian setting. The ideas are as follows.

###### Sketch of the proof of Theorem 1.1.

Firstly, we choose a $C^{0}$-dense function $\\{\psi_{i}\\}_{i\geq 1}$ in
$C^{\infty}(\Sigma,\mathbb{R})$. By using the volume property of spectral
invariants, i.e. Proposition 2.1 (5), we have a sequence of local lipschitz
maps

$\gamma_{k}(\cdot):\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})\rightarrow\mathbb{R}(\lambda\mapsto\frac{c_{\varsigma_{k}}(S^{*}_{F}\Sigma,\lambda)}{\sqrt{2k}}-\sqrt{\mathrm{vol}(S^{*}_{F}\Sigma,\lambda)}).$

which converges to 0 as $k\rightarrow+\infty$. For any
$\lambda_{0}\in\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$ and any open
neighborhood $\mathcal{U}\subset\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$ of
$\lambda_{0}$, we further choose a family of contact forms
$\lambda_{\tau\in[0,1]^{N}}$ in $\mathcal{U}$, then the following composition
becomes a sequence of lipschitz maps

$\gamma_{k}^{\prime}:[0,1]^{N}\rightarrow\mathbb{R}\
(\tau\mapsto\gamma_{k}(\lambda_{\tau})),$

which converges to 0 uniformly as $k\rightarrow+\infty$. Moreover, if there
exists a $\tau$ such that $\lambda_{\tau}$ satisfies
$\frac{\partial\lambda_{\tau}}{\partial\tau_{i}}=\frac{\psi_{i}}{\hat{c}}\lambda_{\tau}$
and $\nabla\gamma_{k}^{\prime}$ is differentiable at $\tau$ with nondegenerate
$\lambda_{\tau}$, then by Proposition 2.3, the differential of
$\gamma_{k}^{\prime}(\tau)$ satisfies

(1.10)
$\frac{\partial\gamma_{k}^{\prime}(\tau)}{\partial\tau_{i}}\sqrt{\mathrm{vol}(S^{*}_{F}\Sigma,\lambda_{\tau})}=\frac{1}{\hat{c}}(\sqrt{\frac{\mathrm{vol}(S^{*}_{F}\Sigma,\lambda_{\tau})}{2k}}C_{\lambda_{\tau}}(\psi_{i}\lambda_{\tau})-\int_{S^{*}_{F}\Sigma}\psi_{i}\lambda_{\tau}\wedge
d\lambda_{\tau}),$

for some Reeb 1-current $C_{\lambda_{\tau}}\in
L(S^{*}_{F}\Sigma,\lambda_{\tau})$. If there exists a common contact form
$\lambda^{\prime}=\lambda_{\tau_{N}}$ for any $N>0$ such that
$\tau_{N}\in[0,1]^{N}$ is a critical point of $\gamma_{k_{N}}^{\prime}$ for
some $k_{N}$, then (1.10) implies that the Reeb 1-current

$C_{N}=(2k_{N})^{-\frac{1}{2}}\mathrm{vol}(S^{*}_{F}\Sigma,\lambda^{\prime})^{\frac{1}{2}}C_{\lambda^{\prime}}$

in $L(S^{*}_{F}\Sigma,\lambda^{\prime})$ satisfies

$C_{N}(\psi_{i}\lambda^{\prime})=\int_{S^{*}_{F}\Sigma}\psi_{i}\lambda^{\prime}\wedge
d\lambda^{\prime},\ \forall i=1,\cdots,N.$

Therefore, the sequence $\\{C_{N}\\}_{N\geq 1}$ are what we need in (1.6) with
respect to $\lambda^{\prime}$ and the arbitrariness of $\lambda_{0}$ and
$\mathcal{U}$ provide the genericity of $\lambda^{\prime}$.

Secondly, to obtain $\\{C_{N}\\}_{N\geq 1}$ for $\lambda^{\prime}$, we need
the family $\lambda_{\tau\in[0,1]^{N}}$, which satisfies that $\lambda_{\tau}$
is nondegenerate at $\tau^{j,m}_{N}$, where $\tau^{j,m}_{N}$ are some
sequences to play the role of $\tau_{N}$, since the lipschitz function
$\gamma_{k_{N}}^{\prime}$ may not possess a critical point. To be specific,
one can choose $\tau^{j,m}_{N}$ such that $\gamma_{k_{N}}^{\prime}$ is
differentiable at $\tau^{j,m}_{N}$ with smaller and smaller gradient as
$N\rightarrow+\infty$. This idea comes from the proof of Marques, Neves and
Song in [23]. To find such a family $\lambda_{[0,1]^{N}}$, we will follow the
steps of Irie in [17], and it turns out that such families exists generically
(see. Lemma 3.3), which is similar to Lemma 3.2 in [17]. Compare to the proof
in [17], the main difficulties here are the transversality arguments (see
Claim 1 and Claim 2), since we are working with the subspace
$\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$ (resp.
$\Lambda^{sFin}(S^{*}_{F}\Sigma,\xi_{F})$) of
$\Lambda(S^{*}_{F}\Sigma,\xi_{F})$, which denotes the collection of all
contact forms on $(S^{*}_{F}\Sigma,\xi_{F})$.

Finally, for generic
$\lambda^{\prime}\in\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$, once we obtain
$C_{N}\in L(S^{*}_{F}\Sigma,\lambda^{\prime})$ from (1.10) and the arguments
above such that

$|C_{N}(\psi_{i}\lambda^{\prime})-\int_{S^{*}_{F}\Sigma}\psi_{i}\lambda^{\prime}\wedge
d\lambda^{\prime}|<\frac{1}{N},\ \forall i=1,\cdots,N,$

(see Claim 4), we can further perturb the corresponding Finsler metric
$F_{\lambda^{\prime}}$ locally such that all the Reeb orbits in $C_{N}$ are
nondegenerate. This is similar to the Lemma 3.1 in [17], however, the main
difference is that the perturbation of $\lambda^{\prime}$ must be in subspace
$\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$ rather than
$\Lambda(S^{*}_{F}\Sigma,\xi_{F})$. To this point, we construct such
perturbations of $\lambda^{\prime}$ specifically, and discuss the variation of
linearized Poincaré map under such a restricted perturbation (see Lemma 3.1).
∎

This paper is organized as follows: In Section 2, we firstly introduce the
well known results of ECH spectral invariants and then list some useful
relations between the Reeb flow and Finsler (Riemannian) geodesic flow. In
Section 3, we mainly prove Lemma 3.2 and 3.3 for generic Finsler (Riemannian)
metrics by using the transversal techniques. In Section 4, we complete the
proofs from Theorem 1.1 to Theorem 1.4 by using the volume property of ECH
spectral invariants, genericity results in Section 3 and the relations between
Reeb flow and geodesic flow. In section 5, some useful results will be listed
as an appendix.

##  2\. Preparation

In this section, at first, we briefly introduce Hutchings’ embedded contact
homology (ECH), especially, the volume property of ECH spectral invariants and
also the Lipschitz property and differentiability of ECH spectral invariants
of Irie in [17], which are the main machineries used in our proofs. Then we
list some useful related facts between the Finsler (Riemannian) geodesic flow
and Reeb flow. For more details of them, we refer to [9], [15], [10], [11] and
the references therein.

###  2.1. Spectral invariant of embedded contact homology

Let $(Y,\xi=\ker\lambda)$ be an oriented closed contact 3-dimensional manifold
with contact structure $\xi$ and contact form $\lambda$, which satisfies
$\lambda\wedge d\lambda>0$ everywhere on $Y$. Each contact form $\lambda$
induces a so-called Reeb vector field $R_{\lambda}$, which is uniquely
determined by $\lambda(R_{\lambda})=1$ and $R_{\lambda}\in\ker d\lambda$. Let
$\varphi_{\lambda}^{t}$ be the corresponding flow of $R_{\lambda}$, then the
periodic orbits of $R_{\lambda}$ are so-called Reeb orbits of $\lambda$ and we
denote by $R(Y,\lambda)$ the collection of all Reeb orbits of $\lambda$. If
$\gamma\in R(Y,\lambda)$ is a $T$-periodic Reeb orbit of $\lambda$, then the
restriction of linearized time $T$ flow

(2.1)
$P_{\gamma}=d(\varphi^{T}_{\lambda})|_{\xi_{\gamma(0)}}:\xi_{\gamma(0)}\rightarrow\xi_{\gamma(T)}$

is called the linearized Poincaré map and it is symplectic with respect to
$d\lambda$.

Let $c_{1}(\xi)$ be the first Chern class of $(\xi,J)$ with a complex
structure $J$ such that $d\lambda_{x}(\cdot,J_{x}\cdot)>0$ for any $x\in Y$
and write $\mathbb{Z}_{d}=\mathbb{Z}/d\mathbb{Z}$. For any $\Gamma\in
H_{1}(Y,\mathbb{Z})$, one can define a $\mathbb{Z}_{2}$-vector space
$\mathrm{ECC}_{*}(Y,\lambda,\Gamma)$ generated by ECH generators with a
relative $\mathbb{Z}_{d}$-grading, where an ECH generator $\alpha$ has the
form $\alpha=\sum_{i=1}^{k}m_{i}\gamma_{i}\in L_{\mathbb{Z}_{+}}(Y,\lambda)$
with $[\alpha]=\Gamma$ and $m_{i}=1$ whenever $P_{\gamma_{i}}$ has eigenvalues
in $\mathbb{R}\setminus\\{\pm 1\\}$ and $d$ denotes the divisibility of
$c_{1}(\xi)+2\mathrm{PD}(\Gamma)\in H^{2}(Y,\mathbb{Z})$. In particular, there
always exists a $\Gamma$ such that $c_{1}(\xi)+2\mathrm{PD}(\Gamma)\in
H^{2}(Y,\mathbb{Z})$ is torsion. We fix such $\Gamma$ from now on, then
$\mathrm{ECC}_{*}(Y,\lambda,\Gamma)$ has a relative $\mathbb{Z}$-grading and
one can define the homology of $\mathrm{ECC}_{*}(Y,\lambda,\Gamma)$ with a
differential, i.e., $\mathrm{ECH}_{*}(Y,\lambda,\Gamma)$. By Taubes [27], the
homology is independent of $\lambda$, then we can also write
$\mathrm{ECH}_{*}(Y,\xi,\Gamma)$.

Let $\mathcal{A}:L(Y,\lambda)\cap\Gamma\rightarrow\mathbb{R}_{\geq
0}(\alpha=\sum_{i=1}^{k}a_{i}\gamma_{i}\mapsto\int_{\alpha}\lambda=\sum_{i=1}^{k}a_{i}T_{i})$,
where $T_{i}$ is the period of $\gamma_{i}$. Let
$\mathrm{ECC}^{a}_{*}(Y,\lambda,\Gamma)$ be the subcomplex of
$\mathrm{ECC}_{*}(Y,\lambda,\Gamma)$, which is generated by ECH generators
$\alpha$ with $\mathcal{A}(\alpha)<a$, then we can define the filtered ECH to
be the homology of this subcomplex, which we denote by
$\mathrm{ECH}^{a}_{*}(Y,\lambda,\Gamma)$. Then for any element
$\varsigma\in\mathrm{ECH}_{*}(Y,\lambda,\Gamma)\setminus\\{0\\}$, one can
defined a spectral invariant

(2.2) $c_{\varsigma}(Y,\lambda)=\inf\\{a\in\mathbb{R}_{+}|\
\varsigma\in\mathrm{Im}\
i^{a}:\mathrm{ECH}^{a}_{*}(Y,\lambda,\Gamma)\rightarrow\mathrm{ECH}_{*}(Y,\xi,\Gamma)\\}.$

######  Proposition 2.1.

(c.f. Section 2 in [17]) The spectral invariant satisfies:

  * (1)

If $c_{\varsigma}(Y,\lambda)>0$, then there exists $\alpha\in
L_{\mathbb{Z}_{+}}(Y,\lambda)$ such that
$c_{\varsigma}(Y,\lambda)=\mathcal{A}(\alpha)$.

  * (2)

$c_{\varsigma}(Y,a\lambda)=ac_{\varsigma}(Y,\lambda),\ \forall a>0$.

  * (3)

$c_{\varsigma}(Y,\lambda)\leq c_{\varsigma}(Y,f\lambda)$ for any $f\in
C^{\infty}(Y,\mathbb{R}_{\geq 1})$.

  * (4)

$\displaystyle\lim_{i\rightarrow+\infty}c_{\varsigma}(Y,f_{i}\lambda)=c_{\varsigma}(Y,\lambda)$
if $\displaystyle\lim_{i\rightarrow+\infty}f_{i}=1$ in $C^{0}$ topology.

  * (5)

There exists a sequence $\\{\varsigma_{k}\\}_{k\geq
1}\subset\mathrm{ECH}_{*}(Y,\xi,\Gamma)$ such that

$\displaystyle I(\varsigma_{k+1},\varsigma_{k})=2\
\&\lim_{k\rightarrow+\infty}\frac{c_{\varsigma_{k}}(Y,\lambda)}{\sqrt{2k}}=\sqrt{\mathrm{vol(Y,\lambda)}},\
\mathrm{vol(Y,\lambda)}=\int_{Y}\lambda\wedge d\lambda.$

As a conclusion, we have the following results, which are proved in [17].

######  Proposition 2.2.

(c.f. Lemma 3.7 in [17]) Let
$\gamma_{k}(\lambda):=\frac{c_{\varsigma_{k}}(Y,\lambda)}{\sqrt{2k}}-\sqrt{\mathrm{vol}(Y,\lambda)},\
k\geq 1$. Then there exists $c(\lambda)\in\mathbb{R}_{+}$ such that

$|\gamma_{k}(e^{f_{1}}\lambda)-\gamma_{k}(e^{f_{2}}\lambda)|\leq
c(\lambda)\|f_{1}-f_{2}\|_{C^{0}},\ \forall f_{1,2}\in C^{\infty}(Y,[-1,1]),\
k\geq 1.$

######  Proposition 2.3.

(c.f. Lemma 3.4 in [17]) Let $\lambda_{\tau\in[0,1]^{N}}$ be a smooth
$[0,1]^{N}$-family of contact forms in $(Y,\xi)$ and
$\varsigma\in\mathrm{ECH}(Y,\xi,\Gamma)\setminus\\{0\\}$. If
$\lambda_{\tau_{0}}$ is nondegenerate, then

  * (1)

There exists $C\in L_{\mathbb{Z}_{+}}(Y,\lambda_{\tau_{0}})$ such that
$C(\lambda_{\tau_{0}})=c_{\varsigma}(Y,\lambda_{\tau_{0}})$.

  * (2)

Assume $\tau\mapsto c_{\varsigma}(Y,\lambda_{\tau})$ is differentiable at
$\tau_{0}$, then

$C(\frac{\partial\lambda_{\tau}}{\partial\tau_{i}}|_{\tau_{0}})=\frac{\partial
c_{\varsigma}(Y,\lambda_{\tau})}{\partial\tau_{i}}|_{\tau=\tau_{0}},\
\tau=(\tau_{1},\cdots,\tau_{N}).$

###  2.2. Geodesic flows and Reeb flows

Let $(\Sigma,F)$ be a Finsler surface with $F$ satisfying (I) to (III) in
Section 1. Since $F|_{x}(v)^{2}:=F(x,v)^{2}$ is convex with respect to $v$ by
(III), therefore, by inverse function theorem, the Legendrian transformation
$\mathcal{L}_{F}:T\Sigma\rightarrow T^{*}\Sigma$ defined in (1.2) is one to
one corresponding to each other. Furthermore, by condition (II), we have
$aF(v_{1},v_{2})=F(av_{1},av_{2})(a>0)$ for any
$v=v_{1}\frac{\partial}{\partial q_{1}}+v_{2}\frac{\partial}{\partial
q_{2}}\in T_{q}\Sigma$. By differentiating $v_{i}$ on both sides, we obtain
that

$\frac{\partial F}{\partial v_{i}}(av)=\frac{\partial F}{\partial v_{i}}(v),\
\forall a>0,\ i=1,2,$

then we have

(2.3) $\mathcal{L}_{F}(av)=\sum_{i=1}^{2}F(av)\frac{\partial F}{\partial
v_{i}}(av)dq_{i}=a\mathcal{L}_{F}(v),\ \forall a>0.$

If $F\in\mathcal{F}^{s}(\Sigma)$, then (2.3) is true for any $a\neq 0$, since
$F(av_{1},av_{2})=-aF(v_{1},v_{2})(a<0)$, by differentiating $v_{i}$ on both
sides, we have $\frac{\partial F}{\partial v_{i}}(av)=-\frac{\partial
F}{\partial v_{i}}(v),\forall a<0$. Then we can introduce a dual norm
$F^{*}(\mathcal{L}_{F}(\cdot))=F(\cdot)$, which satisfies (I) to (III) and
(II’) on $T^{*}\Sigma$ by definition (1.2) and the property of Legendrian
transformation. In particular, (III) implies that $S^{*}_{F}\Sigma$ is
$F^{*}$-unit cotangent bundle with convex fiber and symmetric convex fiber if
$F\in\mathcal{F}^{s}(\Sigma)$. Therefore, under the local coordinates
$(q,p)=p_{1}dq_{1}+p_{2}dq_{2}$ on $T^{*}\Sigma$, the Liouville vector field
$X=p\frac{\partial}{\partial p}$ determined by
$\iota_{X}\omega_{can}=\lambda_{can}$ transverses to $S^{*}_{F}\Sigma$ and
$(S^{*}_{F}\Sigma,\xi_{F}=\ker\lambda(F))$ forms a contact manifold, where
$\lambda(F):=\lambda_{can}|_{S^{*}_{F}\Sigma}$ and
$\omega_{can}=d\lambda_{can}$, which can be written as $pdq|_{F^{*}(p)=1}$ and
$dp\wedge dq|_{F^{*}(p)=1}$ locally.

To measure the length of closed geodesics, we let

(2.4) $\beta=\frac{\partial F}{\partial v_{1}}dq_{1}+\frac{\partial
F}{\partial v_{2}}dq_{2}\ (\mathrm{resp.}\ \beta(F)=\beta|_{S_{F}\Sigma})$

be a 1-form on $T\Sigma$ (resp. $S_{F}\Sigma$). One can easily check that
$\beta$ is independent of the choice of $(q_{1},q_{2})$. Then $\beta$ is
global defined which is socalled Hilbert form in [11]. By Proposition 2.1 and
2.2 in [11], $(S_{F}\Sigma,\beta(F))$ is also a contact manifold. For any
$\sigma\in\mathrm{CG}(\Sigma,F)$, $\tilde{\sigma}=(\sigma,\dot{\sigma})$ in
$T\Sigma$, the length of $\sigma$ is

(2.5)
$l_{F}(\sigma)=\int_{0}^{l_{F}(\sigma)}F(\dot{\sigma}(t))dt=\int_{0}^{l_{F}(\sigma)}(\nabla(F|_{\sigma(t)})(\dot{\sigma}(t)),\dot{\sigma}(t))dt=\int_{\tilde{\sigma}}\beta(F),$

where the second equality follows from $F(v)=(\nabla(F|_{q})(av),v),\forall
v\in T_{q}\Sigma,a>0$.

To measure the volume of $(\Sigma,F)$, we define the volume form on
$(\Sigma,F)$ by

(2.6)
$d\mathrm{vol}_{F}(q)=\frac{1}{\pi}\left(\int_{D^{*}_{F,q}\Sigma}dp_{1}\wedge
dp_{2}\right)dq_{1}\wedge dq_{2},$

where $D^{*}_{F,q}\Sigma$ is the unit disk bounded by the fiber
$S^{*}_{F,q}\Sigma$ of $S^{*}_{F}\Sigma$ at $q$. Since $\omega_{can}$ is
independent of the choice of $(q,p)$, so is $dp_{1}\wedge dp_{2}\wedge
dq_{1}\wedge dq_{2}$. Therefore, $d\mathrm{vol}_{F}$ is globally well-defined
and we denote by

$d\mathrm{vol}_{F}(f):=\int_{\Sigma}f(q)d\mathrm{vol}_{F}(q),\ \forall f\in
C^{\infty}(\Sigma,\mathbb{R}).$

Especially, we have $\mathrm{vol}(\Sigma,F):=d\mathrm{vol}_{F}(1)$.

If $F_{g}^{2}(\cdot)=g(\cdot,\cdot)\in\mathcal{M}(\Sigma)$, then
$S^{*}_{g}\Sigma=S^{*}_{F_{g}}\Sigma$ is the $g^{*}$-unit cotangent bundle
with dual normal $\|\mathcal{L}_{g}(\cdot)\|_{g^{*}}=\|\cdot\|_{g}$, where
$\mathcal{L}_{g}(\cdot):=\mathcal{L}_{F_{g}}(\cdot)=(v,\cdot)_{g}$ by
definition (1.2). More precisely, under the local coordinates $(q,p)$ and for
any $v=v_{1}\frac{\partial}{\partial q_{1}}+v_{2}\frac{\partial}{\partial
q_{2}}$ in $T\Sigma$, we have

$p:=\mathcal{L}_{F_{g}}(v)=p_{1}dq_{1}+p_{2}dq_{2},\
p_{i}=g_{i1}v_{1}+g_{i2}v_{2},\ i=1,2,$

and $\beta(F_{g})|_{v}(\cdot)=(v,\cdot)_{g}$, where
$g_{ij}=g(\frac{\partial}{\partial q_{i}},\frac{\partial}{\partial q_{j}})$.
Moreover, write $G=(g_{ij})_{2\times 2},\ G^{*}=(g_{ij}^{*})_{2\times 2},\
\bar{v}=(v_{1},v_{2})^{T},\ \bar{p}=(p_{1},p_{2})^{T}=G\bar{v}$ and we have

$(G\bar{v},\bar{v})=(v,v)_{g}=(p,p)_{g^{*}}=(G^{*}\bar{p},\bar{p})=(G^{*}G\bar{v},G\bar{v}),$

then $G^{*}=G^{-1}$. Since

(2.7)
$\int_{D^{*}_{g,q}\Sigma}dp_{1}dp_{2}=\mathrm{Area}\\{\bar{p}\in\mathbb{R}^{2}|\
(G^{*}\bar{p},\bar{p})=1\\}=\pi\det(G)^{\frac{1}{2}},$

then
$d\mathrm{vol}_{g}(q):=d\mathrm{vol}_{F_{g}}(q)=\det(G)^{\frac{1}{2}}dq_{1}\wedge
dq_{2}$ is the canonical volume form.

######  Proposition 2.4.

For any $F,F_{1}\in\mathcal{F}(\Sigma)$, we have

  * (1)

$(S^{*}_{F}\Sigma,{F_{1}^{*}(\cdot)}^{-1}\lambda(F))\cong(S^{*}_{F_{1}}\Sigma,\lambda(F_{1}))((q,p)\mapsto(q,{F^{*}_{1}(p)}^{-1}p))$.

  * (2)

For any $f\in C^{\infty}(\Sigma,\mathbb{R}_{+})$, we have

$(S^{*}_{F}\Sigma,(f\circ\pi)\lambda(F))\cong(S^{*}_{fF}\Sigma,\lambda(fF))\
((q,p)\mapsto(q,((fF)^{*}(p))^{-1}p)).$

  * (3)

The Hamiltonian flow $\varphi^{t}_{F^{*}}$ of $F^{*}(\cdot)$ coincides with
the Reeb flow $\varphi^{t}_{\lambda(F)}$ and conjugate to the geodesic flow
$\varphi_{F}^{t}$ on unit tangent bundle $S_{F}\Sigma$ via $\mathcal{L}_{F}$.

  * (4)

The projection $R(S^{*}_{F}\Sigma,\lambda(F))\rightarrow\mathrm{CG}(\Sigma,F)\
(\gamma_{\sigma}\mapsto\sigma=\pi(\gamma_{\sigma}))$ is a one to one
correspondence.

  * (5)

$\gamma_{\sigma}(f\lambda(F))=\sigma(f\beta(F))$ for any
$\gamma\in\mathrm{CG}(\Sigma,F)$ and $f\in C^{\infty}(\Sigma,\mathbb{R})$.

  * (6)

The period $T_{\gamma_{\sigma}}$ of $\gamma_{\sigma}\in
R(S^{*}_{F}\Sigma,\lambda(F))$ coincides with the length $l_{F}(\sigma)$.

  * (7)

$d\lambda(F)(f\lambda(F))=2\pi d\mathrm{vol}_{F}(f)$ for any $f\in
C^{\infty}(\Sigma,\mathbb{R})$.

  * (8)

The volume
$\mathrm{vol}(S^{*}_{F}\Sigma,\lambda(F))=2\pi\mathrm{vol}(\Sigma,F)$.

###### Proof.

To show (1). Since $\lambda(F_{1})=pdq|_{F_{1}^{*}(p)=1}$, then the pull back
of $\lambda(F_{1})$ via $\phi:p\mapsto F_{1}^{*}(p)^{-1}p$ becomes

$\phi^{*}\lambda(F_{1})=F_{1}^{*}(p)^{-1}pdq|_{F^{*}(p)=1}=F_{1}^{*}(p)^{-1}\lambda(F).$

To show (2). Let $F_{1}=fF$. By definition (1.2), we obtain

(2.8) $p:=\mathcal{L}_{F_{1}}(v)=F_{1}(v)\frac{\partial F_{1}}{\partial
v}(v)=f(q)^{2}F(v)\frac{\partial F}{\partial v}(v).$

Then condition (II) and (2.8) implies that for any $(q,p)\in S^{*}_{F}\Sigma$,

$\displaystyle F_{1}^{*}(p)$
$\displaystyle=F_{1}(v)=f(q)F(v)=f(q)F^{*}(\mathcal{L}_{F}(v))=f(q)F^{*}(F(v)\frac{\partial
F}{\partial v}(v))$ $\displaystyle=f(q)^{-1}F^{*}(f(q)^{2}F(v)\frac{\partial
F}{\partial v}(v))=f(q)^{-1}F^{*}(p)=f(q)^{-1}.$

Hence, (2) follows from (1).

To show (3). Firstly, we let $X_{F^{*}}$ be the Hamiltonian vector field of
$F^{*}$ on $S^{*}_{F}\Sigma$. Since
$F^{*}|_{q}(ap)=F|_{q}(\mathcal{L}_{F}^{-1}(ap))=aF^{*}|_{q}(p),\ \forall
a>0$, which follows from (2.3), then for any $(q,p)\in S^{*}_{F}\Sigma$, we
have $d(F^{*}|_{q})(ap)(p\frac{\partial}{\partial p})=F^{*}|_{q}(p),\ \forall
a>0$, and then

$\lambda(F)(X_{F^{*}})=\omega_{can}(p\frac{\partial}{\partial
p},X_{F^{*}})=dF^{*}(p)(p\frac{\partial}{\partial p})=F^{*}(p)=1$

and $\omega_{can}(\cdot,X_{F^{*}})=dF^{*}(\cdot)=0$ on $S^{*}_{F}\Sigma$.
Therefore, $X_{F^{*}}$ is the Reeb vector field $R_{\lambda(F)}$ of
$\lambda(F)$. Secondly, let $\sigma\in\mathrm{CG}(\Sigma,F)$ and write
$\tilde{\sigma}=(\sigma,\dot{\sigma})$ in $S_{F}\Sigma$. Then by homogeneity
of $F$ and definition of Hilbert form $\beta$ above, we have

$\beta(F)(\tilde{\sigma})=(\nabla(F|_{\sigma})(\dot{\sigma}),\dot{\sigma})=F(\dot{\sigma})=1.$

Since $\sigma$ is also a critical point of length function $l_{F}(\cdot)$,
therefore, by (2.5), for any $\sigma_{s}:[0,l_{F}(\sigma)]\rightarrow\Sigma,\
s\in(-\epsilon,\epsilon)$, with $\sigma_{0}=\sigma$ and
$\frac{d\sigma_{s}}{ds}|_{s=0}=\eta$, we have

$0=\frac{dl_{F}(\sigma_{s})}{ds}|_{s=0}=\frac{d}{ds}|_{s=0}\int_{\tilde{\sigma}_{s}}\beta(F)=\int_{\sigma}\mathcal{L}_{\eta}\beta(F)=\int_{\sigma}\iota_{\eta}d\beta(F),$

where $\mathcal{L}_{\eta}$ is the Lie derivative. Since $\eta$ can be any
vector field along $\sigma$, therefore, $\dot{\sigma}$ is in $\ker(d\beta(F))$
and $\tilde{\sigma}$ becomes a periodic orbit of $R_{\beta(F)}$, the Reeb
vector field of $\beta(F)$. Finally, by (1.2), the pull back

$\mathcal{L}_{F}^{*}\lambda(F)=F(v)\frac{\partial F(v)}{\partial
v}dq=\frac{\partial F(v)}{\partial v}dq=\beta(F),\ \forall(q,v)\in
S_{F}\Sigma,$

which means that the $(d\mathcal{L}_{F})R_{\beta(F)}=R_{\lambda(F)}$.
Therefore, $\mathcal{L}_{F}(\tilde{\sigma})\in R(S^{*}_{F}\Sigma,\lambda(F))$
and (3), (4) follows.

To show (5). Since $\gamma_{\sigma}=\mathcal{L}_{F}(\tilde{\sigma})$, by
(1.5), we have

$\displaystyle\gamma_{\sigma}(f\lambda(F))=\int_{\gamma_{\sigma}}f\lambda(F)=\int_{\tilde{\sigma}}(f\circ\mathcal{L}_{F})\mathcal{L}_{F}^{*}\lambda(F)=\int_{\tilde{\sigma}}f\beta(F)=\sigma(f\beta(F)).$

When $f\equiv 1$, we have (6), i.e. $T_{\gamma_{\sigma}}=l_{F}(\sigma)$.

To show (7). Denote by $S^{*}_{F,q}\Sigma$ the fiber of $S^{*}_{F}\Sigma$ at
$q$, which is bounding the disk $D^{*}_{F,q}\Sigma\subset T^{*}_{q}\Sigma$. By
definition (2.6), we obtain

$\displaystyle d\lambda(F)(f\lambda(F))$
$\displaystyle=\int_{S^{*}_{F}\Sigma}f\lambda(F)\wedge
d\lambda(F)=\int_{\Sigma}f(q)(\int_{S^{*}_{F,q}\Sigma}p_{1}dp_{2}-p_{2}dp_{1})(dq_{1}\wedge
dq_{2})$
$\displaystyle=2\int_{\Sigma}f(q)(\int_{D^{*}_{F,q}\Sigma}dp_{1}dp_{2})dq_{1}\wedge
dq_{2}=2\pi\int_{\Sigma}f(q)d\mathrm{vol}_{F}(q)$ $\displaystyle=2\pi
d\mathrm{vol}_{F}(f).$

(8) is the conclusion of (7), when $f\equiv 1$. ∎

As a conclusion, we can obtain the following results, similarly.

######  Proposition 2.5.

For any $g,g_{1}\in\mathcal{M}(\Sigma)$, we have

  * (1)

$(S^{*}_{g}\Sigma,\|p\|^{-1}_{g_{1}^{*}}\lambda(g))\cong(S^{*}_{g_{1}}\Sigma,\lambda(g_{1}))((q,p)\mapsto(q,\|p\|^{-1}_{g_{1}^{*}}p))$.

  * (2)

For any $f\in C^{\infty}(\Sigma,\mathbb{R}_{+})$, we have

$(S^{*}_{g}\Sigma,(f\circ\pi)\lambda(g))\cong(S^{*}_{f^{2}g}\Sigma,\lambda(f^{2}g))\
((q,p)\mapsto(q,\|p\|^{-1}_{(f^{2}g)^{*}}p)).$

  * (3)

The Hamiltonian flow $\varphi^{t}_{g^{*}}$ of $\|\cdot\|_{g^{*}}$ coincides
with the Reeb flow $\varphi^{t}_{\lambda(g)}$ and conjugate to the geodesic
flow $\varphi_{g}^{t}$ on unit tangent bundle $S_{g}\Sigma$ via
$\mathcal{L}_{g}$.

  * (4)

The projection $R(S^{*}_{g}\Sigma,\lambda(g))\rightarrow\mathrm{CG}(\Sigma,g)\
(\gamma_{\sigma}\mapsto\sigma)$ is a one to one correspondence.

  * (5)

$\gamma_{\sigma}(f\lambda(g))=\sigma(fds)$ for any
$\sigma\in\mathrm{CG}(\Sigma,g)$ and $f\in C^{\infty}(\Sigma,\mathbb{R})$.

  * (6)

The period $T_{\gamma_{\sigma}}$ of $\gamma_{\sigma}\in
R(S^{*}_{g}\Sigma,\lambda(g))$ coincides with the length $l_{g}(\sigma)$.

  * (7)

$d\lambda(g)(f\lambda(g))=2\pi d\mathrm{vol}_{g}(f)$ for any $f\in
C^{\infty}(\Sigma,\mathbb{R})$.

  * (8)

The volume
$\mathrm{vol}(S^{*}_{g}\Sigma,\lambda(g))=2\pi\mathrm{vol}(\Sigma,g)$.

###### Proof.

To show (1). Since $\lambda(g_{1})=pdq|_{\|p\|_{g_{1}^{*}}=1}$, then the pull
back of $\lambda(g)$ via $\phi:p\mapsto\|p\|^{-1}_{g_{1}^{*}}p$ becomes
$\phi^{*}\lambda(g_{1})=\|p\|^{-1}_{g_{1}^{*}}pdq|_{\|p\|_{{g}^{*}}=1}=\|p\|^{-1}_{g_{1}^{*}}\lambda(g).$

To show (2). Let $g_{1}=f^{2}g$. By definition (1.2), we have

(2.9)
$p:=\mathcal{L}_{f^{2}g}(v)=(v,\cdot)_{f^{2}g}=(f(q)^{2}v,\cdot)_{g}=\mathcal{L}_{g}(f(q)^{2}v)=f(q)^{2}\mathcal{L}_{g}(v).$

Then (2) follows from (1), (2.9) and the fact that

$\displaystyle\|p\|_{(f^{2}g)^{*}}$
$\displaystyle=\|\mathcal{L}_{f^{2}g}(v)\|_{(f^{2}g)^{*}}=\|v\|_{f^{2}g}=f(q)\|v\|_{g}=f(q)^{-1}\|f(q)^{2}\mathcal{L}_{g}(v)\|_{g^{*}}$
$\displaystyle=f(q)^{-1}\|p\|_{{g}^{*}}=f(q)^{-1}.$

To show (3). Firstly, the Hamiltonian vector field of $\|\cdot\|_{g^{*}}$ is
$X_{g^{*}}=p\frac{\partial}{\partial q}|_{\|p\|_{g^{*}}=1}$, since
$pdp=\omega_{can}(\cdot,p\frac{\partial}{\partial q})$. Meanwhile, $X_{g^{*}}$
is also the Reeb vector field of $\lambda(g)$, since $\lambda(g)(X_{g^{*}})=1$
and $pdp=\omega_{can}(\cdot,X_{g^{*}})=0$ on $S^{*}_{g}\Sigma$. Therefore,
$\varphi^{t}_{g^{*}}=\varphi^{t}_{\lambda(g)}$ and (3) follows. To the direct
proof of $\phi^{t}_{g^{*}}\circ\mathcal{L}_{g}=\phi^{t}_{g}$, we refer to
Theorem 2.3.1 in [12]. (4) is a conclusion of (3).

To show (5). Let $s$ be the arc length parameter of $\sigma$. By definition,
we have

$\displaystyle\gamma_{\sigma}(f\lambda(g))$
$\displaystyle=\int^{T_{\gamma_{\sigma}}}_{0}f(q(t))p(t)(\dot{q}(t))dt=\int^{T_{\gamma_{\sigma}}}_{0}f(q(t))\|\dot{q}(t)\|^{2}_{g}dt$
$\displaystyle\overset{s=t}{=}\int_{\sigma}f(q(s))ds=\sigma(fds).$

When $f\equiv 1$, we have (6), i.e. $T_{\gamma_{\sigma}}=l_{g}(\sigma)$.

To show (7). Denote by $S^{*}_{g,q}\Sigma$ the fiber of $S^{*}_{g}\Sigma$ at
$q$ with bounding the disk $D^{*}_{g,q}\Sigma$. Then by (2.7), we can conclude
that

$\displaystyle d\lambda(g)(f\lambda(g))$ $\displaystyle=$
$\displaystyle\int_{S^{*}_{g}\Sigma}f\lambda(g)\wedge
d\lambda(g)=\int_{\Sigma}f(q)(\int_{S^{*}_{g,q}\Sigma}p_{1}dp_{2}-p_{2}dp_{1})(dq_{1}\wedge
dq_{2})$ $\displaystyle=$ $\displaystyle
2\int_{\Sigma}f(q)(\int_{D^{*}_{g,q}\Sigma}dp_{1}dp_{2})dq_{1}\wedge
dq_{2}=2\pi\int_{\Sigma}f(q)d\mathrm{vol}_{g}(q)$ $\displaystyle=$
$\displaystyle 2\pi d\mathrm{vol}_{g}(f).$

(8) is the conclusion of (7), when $f\equiv 1$. ∎

Let $\sigma\in\mathrm{CG}(\Sigma,F)$, then Proposition2.4 (3) implies that

(2.10) $\gamma_{\sigma}(t)=(\sigma(t),p(t))\in R(S^{*}_{F}\Sigma,\lambda(F)),\
\forall t\in[0,l_{F}(\sigma)],$

where $p(t)(\cdot)=\mathcal{L}_{F}(\dot{\sigma}(t))$. By definition (2.1), we
obtain a linear Poincaré map

(2.11)
$P_{\sigma}:=P_{\gamma_{\sigma}}:\xi_{F}|_{\gamma_{\sigma}(0)}\rightarrow\xi_{F}|_{\gamma_{\sigma}(l_{F}(\sigma))},$

which is symplectic with respect to the symplectic form
$d\lambda(F)|_{\xi_{F}}$. The closed geodesic $\sigma$ is called nondegenerate
if $1\notin\sigma(P_{\sigma})$, and we call $F\in\mathcal{F}(\Sigma)$ bumpy if
$\mathrm{CG}(\Sigma,F)$ are all nondegenerate. Similarly, on a contact
manifold $(Y,\xi=\ker\lambda)$, a Reeb orbit $\gamma\in R(Y,\lambda)$ is
called nondegenerate if $1\notin\sigma(P_{\gamma})$. If $R(Y,\lambda)$ are all
nondegenerate, then we call $\lambda$ nondegenerate. We also call a Finsler
1-current $\sum_{i=1}^{k}a_{i}\sigma_{i}\in l_{\mathrm{CG}}(\Sigma,F)$
nondegenerate if $\sigma_{i}$ are all nondegenerate. Similar to the Reeb
1-current in $L(S^{*}_{F},\lambda(F))$. Therefore, by above definitions,
$\sigma$ and $\gamma_{\sigma}$ are both nondegenerate or not, and $F$ is bumpy
if and only if $\lambda(F)$ is nondegenerate.

##  3\. Generic results for Finsler and Riemannian surfaces

In this section, we are aiming to prove Lemma 3.2 and 3.3. Which is analogous
of Lemma 3.1 and 3.2 in [17]. In another word, on the one hand, any Finsler
(Riemannian) 1-current on $(\Sigma,F)$ can be nondegenerate under an arbitrary
small perturbation of $F$ in $\mathcal{F}(\Sigma)$. On the other hand, For any
$N\in\mathbb{N}_{+}$, we prove that almost all the smooth $[0,1]^{N}$-family
of Finsler metrics are generically bumpy, i.e. nonbumpy Finsler metric has
measure zero in $[0,1]^{N}$. The same conclusions are also available for
symmetric Finsler metrics and Riemannian metrics.

Before we prove the Lemma 3.2, we are going to study the linearized Poincaré
map $P_{\sigma}$ under the perturbation of metrics. Let
$F,F_{1}\in\mathcal{F}(\Sigma)$, then Proposition 2.4 (1) provides the
following isomorphism (or we call strict contactomorphism)

$(S^{*}_{F}\Sigma,F_{1}^{*}(\cdot)^{-1}\lambda(F))\cong(S^{*}_{F_{1}}\Sigma,\lambda(F_{1}))((q,p)\mapsto(q,F_{1}^{*}(p)^{-1}p)).$

Write $f(q,p)=-\ln F_{1}^{*}|_{q}(p)$, then
$e^{f}\lambda(F)\in\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$ with respect to
$F_{1}$. Let $U$ be a sufficient small tubular neighborhood of
$\gamma_{\sigma}\in R(S^{*}_{F}\Sigma,\lambda(F))$ with
$\pi(\gamma_{\sigma})=\sigma\in\mathrm{CG}(\Sigma,F)$. It is well known that,
for any $\gamma\in R(S^{*}_{F}\Sigma,\lambda(F))$, if $f|_{\gamma}=0$ and
$df|_{\gamma}=0$, then we also have $\gamma\in
R(S^{*}_{F}\Sigma,e^{f}\lambda(F))$, since the Reeb vector field does not
change at $\gamma$. Now, for any open subset $\mathcal{U}\subset\Sigma$ with
local coordinate $(q,p)$ in $T^{*}\mathcal{U}:=\pi^{-1}(\mathcal{U})$ and
$S^{*}_{F}\mathcal{U}:=S^{*}_{F}\Sigma\cap T^{*}\mathcal{U}$, we define

(3.1) $\mathcal{F}^{pert}_{\mathcal{U}}:=\\{f\in
C^{\infty}_{0}(S^{*}_{F}\mathcal{U},\mathbb{R})|\
f=-\frac{1}{2}\ln(1+(A(q)p,p)),\ A=A^{T}\\},$

where $A(q)$ is a $2\times 2$ matrix on $T^{*}_{q}\Sigma$ under the
coordinates $p=(p_{1},p_{2})$, and support in $\mathcal{U}$. Then we can
easily obtain that

######  Proposition 3.1.

Assume $f\in\mathcal{F}^{pert}_{\pi(U)}$ and write
$\gamma_{\sigma}(\cdot)=(\sigma(\cdot),p(\cdot))\in
R(S^{*}_{F}\Sigma,\lambda(F))$ as in (2.10), then $f|_{\gamma_{\sigma}}=0,\
df|_{\gamma_{\sigma}}=0$ if and only if

(3.2) $(A(\sigma(t))p(t),p(t))=0\ \&\
((dA|_{\sigma(t)}(v(t))p(t),p(t)),(A(\sigma(t))p(t),V(t)))=0,$

for any $v(t)\in T_{\sigma(t)}\Sigma,\ V(t)\in
T_{p(t)}S^{*}_{F,\sigma(t)}\Sigma$ and $t\in[0,l_{F}(\sigma)]$.

###### Proof.

The proposition follows from the definition (3.1) and the fact that
$T_{\sigma(t)}\Sigma\oplus
T_{p(t)}S^{*}_{F,\sigma(t)}\Sigma=T_{\gamma_{\sigma}(t)}U$. ∎

To construct a variation of $F_{s}\in\mathcal{F}(\Sigma)$ from $F$, we let

(3.3)
$f_{s}(q,p)=-\frac{1}{2}\ln(1+s(\tilde{A}(q)p,p))\in\mathcal{F}^{pert}_{\pi(U)},\
\forall s\in[0,1],$

which can be extended to $T^{*}\pi(U)$ as

(3.4)
$\tilde{f}_{s}(q,p)=-\frac{1}{2}\ln(F^{*}|_{q}(p)^{2}+s(\tilde{A}(q)p,p)),\
F^{*}|_{q}(p):=F^{*}(q,p),$

and satisfies
$e^{\tilde{f}_{s}(q,p)}=(F^{*}|_{q}(p)^{2}+s(\tilde{A}(q)p,p))^{-\frac{1}{2}}=F_{s}^{*}|_{q}(p)^{-1}$,
where

(3.5)
$F^{*}_{s}|_{q}(p)=(F^{*}|_{q}(p)^{2}+s(\tilde{A}(q)p,p))^{\frac{1}{2}},\
\forall(q,p)\in T^{*}\pi(U),\ s\in[0,1],$

and $F_{s}^{*}=F^{*}$ outside $T^{*}\pi(U)$. By choosing $\tilde{A}(\cdot)$
sufficiently small, $F^{*}_{s}|_{q}(p)^{2}$ is still convex with respect to
$p$, then the Finsler conditions (I) to (III) and (II’) hold true on
$T^{*}\Sigma$. Then $F_{s}:=F^{*}_{s}\circ\mathcal{L}^{*}_{F_{s}}$ is the
variation of Finsler metrics with $F_{0}=F$, where
$\mathcal{L}^{*}_{F_{s}}:T^{*}\Sigma\rightarrow
T\Sigma((q,p)\mapsto\sum_{i=1}^{2}F^{*}_{s}|_{q}(p)\frac{\partial
F^{*}_{s}|_{q}(p)}{\partial p_{i}}\frac{\partial}{\partial q_{i}})$ is the
dual transformation of $\mathcal{L}_{F_{s}}$. Especially, if
$F\in\mathcal{F}^{s}(\Sigma)$ or $\mathcal{M}(\Sigma)$, then by (3.5), we also
have $F_{s}\in\mathcal{F}^{s}(\Sigma)$ or $\mathcal{M}(\Sigma)$.

Next, we are going to represent $\tilde{A}(\cdot)$ properly. Let $V$ be the
vertical vector field in $U$, which is tangent to the fiber of
$S^{*}_{F}\Sigma$. Then $\\{V,pdq\\}$ forms a basis of $T^{*}_{q}\Sigma$.
Under which we have a matrization
$\tilde{A}(q,p)=(\tilde{A}_{ij}(q,p))_{2\times 2}$ of $\tilde{A}(q)$.

However, the basis $\\{V,pdq\\}$ also depends on variable $p$. To construct a
family of basis which only depends on $q$, we need to work on a much smaller
neighborhood in $U$. Let $U_{1}\subset U$ be a small disk-like open
neighborhood of $\gamma_{\sigma}(l_{F}(\sigma)t_{0})$ for some
$l_{F}(\sigma)t_{0}\in(0,l_{F}(\sigma))\cap\mathrm{reg}(\sigma)$, where
$\mathrm{reg}(\sigma)\subset[0,l_{F}(\sigma)]$ denotes the regular moments of
$\sigma$, i.e. avoiding self-intersections. Then we choose a family of curves
$\sigma_{s}(t)$ covering $\pi(U_{1})$, which satisfies $\sigma_{0}=\sigma$ and
$W_{1}(t)=\frac{\partial\sigma_{s}(t)}{\partial s}|_{s=0}$ is a normal vector
field along $\sigma$ in $\pi(U_{1})$. We note that

$\xi_{F}|_{\gamma_{\sigma}(t)}=\ker pdq|_{(\sigma(t),p(t))}=\\{V,W\\}(t),$

where $W=-p_{2}\frac{\partial}{\partial q_{1}}+p_{1}\frac{\partial}{\partial
q_{2}}$ is a normal vector field along $\sigma$. After scaling $V$ properly,
we can assume that $\\{V,W\\}$ is a symplectic basis. After a
reparametrization of $\sigma_{s}(t)$, we can also assume $W_{1}(t)=W(t)$ in
$\pi(U_{1})$. Then we have new coordinates $(t_{1},t_{2})$ for each point
$q(t_{1},t_{2})=\sigma_{t_{1}}(t_{2})$ in $\pi(U_{1})$ and a new basis
$\\{V_{t_{1},t_{2}},P_{t_{1},t_{2}}\\}$ for each
$T^{*}_{\sigma_{t_{1}}(t_{2})}\Sigma$, where

(3.6)
$P_{t_{1},t_{2}}=\mathcal{L}_{F}(\frac{\dot{\sigma}_{t_{1}}(t_{2})}{F(\dot{\sigma}_{t_{1}}(t_{2}))})\in
S^{*}_{F}\Sigma,\ V_{t_{1},t_{2}}=V(P_{t_{1},t_{2}})\in
T_{P_{t_{1},t_{2}}}S^{*}_{F,\sigma_{t_{1}}(t_{2})}\Sigma.$

Especially, $P_{0,t}=p(t)$. Under the basis
$\\{V_{t_{1},t_{2}},P_{t_{1},t_{2}}\\}$, we can further rewrite

(3.7)
$\tilde{A}(q(t_{1},t_{2}))=\begin{pmatrix}\tilde{A}_{ij}(t_{1},t_{2})\end{pmatrix}_{2\times
2},\ \forall q\in\pi(U_{1}).$

Since the vector field $\bar{W}(t)=\frac{\partial}{\partial
t_{1}}|_{t_{1}=0}(q(t_{1},t),P_{t_{1},t})$ satisfies $\pi_{*}\bar{W}(t)=W(t)$
along $\sigma(t)$, then $\bar{W}(t)=W(t)+a(t)V(t)\in\xi_{F}|_{\sigma(t)}$ for
some function $a(t)$ and transverses to $V(t)$. Since
$\omega_{can}(V(t),W(t))=1$, $\\{\bar{V}(t),\bar{W}(t)\\}$ forms a new
symplectic basis of $\xi_{F}|_{\gamma_{\sigma}(t)}$, where $\bar{V}(t):=V(t)$.
Let

(3.8) $h(q,p)=\frac{\partial f_{s}}{\partial
s}|_{s=0}(q,p)=-\frac{1}{2}(\tilde{A}(q)p,p),$

then under the basis

(3.9)
$\xi_{F}|_{\gamma_{\sigma}(t)}=\\{\bar{V}(t),\bar{W}(t)\\}=\\{V(t),(\frac{\partial
q(0,t)}{\partial t_{1}},\frac{\partial P_{0,t}}{\partial t_{1}})\\},$

the gradient of $h$ is

(3.10) $\displaystyle\nabla h|_{\xi_{F}}(\gamma_{\sigma}(t))$
$\displaystyle=-((\tilde{A}(\sigma(t))p(t),\bar{V}(t)),\frac{\partial}{\partial
t_{1}}|_{t_{1}=0}\frac{1}{2}(\tilde{A}(\sigma_{t_{1}}(t))P_{t_{1},t},P_{t_{1},t}))$
$\displaystyle=(-\tilde{A}_{12}(0,t),-\frac{1}{2}\frac{\partial\tilde{A}_{22}(0,t)}{\partial
t_{1}}),$

and the Hessian of $h$ is

(3.11)
$\displaystyle\nabla^{2}h|_{\xi_{F}}(\gamma_{\sigma}(t))=\begin{pmatrix}-\tilde{A}_{11}(0,t)&-\frac{\partial}{\partial
t_{1}}\tilde{A}_{12}(0,t)\\\ -\frac{\partial}{\partial
t_{1}}\tilde{A}_{12}(0,t)&-\frac{1}{2}\frac{\partial^{2}}{\partial
t_{1}^{2}}\tilde{A}_{22}(0,t)\end{pmatrix}.$

If $f_{s}|_{\gamma_{\sigma}}=0$ and $df_{s}|_{\gamma_{\sigma}}=0$, then
Proposition 3.1, (3.2) implies that

(3.12)
$\displaystyle\tilde{A}_{22}(0,t)=0=(2\tilde{A}_{12}(0,t),\frac{\partial\tilde{A}_{22}(0,t)}{\partial
t_{1}}),\ \forall t\in[0,l_{F}(\sigma)],$

and $\gamma_{\sigma}\in R(S^{*}_{F}\Sigma,F_{s}^{*}(\cdot)^{-1}\lambda(F))$
for any $s\in[0,1]$.

Now, we consider the linearized Poincaré map of $\gamma_{\sigma}$ under the
variation $e^{f_{s}}\lambda(F)$ with $f_{s}\in\mathcal{F}^{pert}_{\pi(U)}$ and
satisfies $f_{s}|_{\gamma_{\sigma}}=0$ and $df_{s}|_{\gamma_{\sigma}}=0,\
\forall s\in[0,1]$. Firstly, we denote by

$\mathcal{F}^{pert}_{\pi(U),0}:=\\{f\in\mathcal{F}^{pert}_{\pi(U)}|\
f|_{\gamma_{\sigma}}=0,\ df|_{\gamma_{\sigma}}=0\\}.$

Then we define the linearized Poincar$\acute{\mathrm{e}}$ return map of
$\gamma_{\sigma}$ in $(S^{*}_{F}\Sigma,e^{f}\lambda(F))$ as

(3.13)
$P_{\sigma}(\cdot):\mathcal{F}^{pert}_{\pi(U),0}\rightarrow\mathrm{Sp}(\xi_{F}|_{\gamma_{\sigma}(0)},\xi_{F}|_{\gamma_{\sigma}(l_{F}(\sigma))})(f\mapsto
P_{\sigma}(f)).$

By Proposition 5.1 in Appendix, there exists a diffeomorphism with preserving
the contact forms, i.e. strict contactomorphism

(3.14) $\Phi:(S^{1}\times
D^{2}_{\epsilon},H(dz+x_{1}dx_{2}))\cong(U,\lambda(F))(\zeta=(z,x_{1},x_{2})\mapsto(q,p)),$

for some function $H:S^{1}\times D^{2}_{\epsilon}\rightarrow\mathbb{R}_{+}$
with $H|_{\Phi^{-1}(U_{1})}\equiv l_{F}(\sigma)$ and $dH(z,0,0)=0$. Then we
can identify $\xi_{F}$ to $\ker(dz+x_{1}dx_{2})$, which is
$\\{\frac{\partial}{\partial x_{1}},\frac{\partial}{\partial x_{2}}\\}$ at
$\gamma_{\sigma}$, and we can also identify their Reeb flows in $S^{1}\times
D^{2}_{\epsilon}$ and $U$. Especially,
$\gamma_{\sigma}(l_{F}(\sigma)t)=\Phi(t,0,0)$.

By choosing $U_{1}$ such that

(3.15)
$U_{1}\cap\gamma_{\sigma}=\gamma_{\sigma}(l_{F}(\sigma)(t_{0}-\epsilon),l_{F}(\sigma)(t_{0}+\epsilon)),$

we have the following result.

######  Lemma 3.1.

Let $f_{s}\in\mathcal{F}^{pert}_{\pi(U),0}$ be defined as in (3.3) with
initial direction $h$ in (3.8), where $\tilde{A}$ is represented as in (3.7)
and supported in $\pi(U_{1})$. By using the local model $\Phi$ in (3.14), we
have

$\displaystyle
dP_{\sigma}|_{f=0}(h)=P_{\sigma}(0)R_{0}(t_{0})^{-1}(\int^{t_{0}+\epsilon}_{t_{0}-\epsilon}\frac{J\nabla^{2}_{x}\tilde{h}}{2l_{F}(\sigma)}(t,0,0)dt)R_{0}(t_{0}),\
J=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},$

where $\tilde{h}=h\circ\Phi$ and $R_{0}(t)$ is the fundamental solution of
$\dot{R}(t)=-\frac{J\nabla^{2}_{x}H}{l_{F}(\sigma)^{2}}(t,0,0)R(t)$.

###### Proof.

Since $H(dz+x_{1}dx_{2})=\Phi^{*}\lambda(F)$ in $U$, then we have the local
model

(3.16) $\Phi:(S^{1}\times
D^{2}_{\epsilon},H_{s}(dz+x_{1}dx_{2}))\cong(U,e^{f_{s}}\lambda(F)),\
H_{s}=e^{f_{s}\circ\Phi}H.$

In the following proof, we identify $U_{1}$ with $\Phi^{-1}(U_{1})$. By
assumption, we have

(3.17) $\displaystyle\mathrm{supp}f_{s}=\mathrm{supp}h\subset\pi(U_{1}),\
H|_{U_{1}}\equiv l_{F}(\sigma)\equiv H|_{S^{1}\times\\{0\\}},$
$\displaystyle\qquad\quad f_{s}|_{\gamma_{\sigma}}=0,\
df_{s}|_{\gamma_{\sigma}}=0=dH|_{S^{1}\times\\{0\\}}.$

Then we can also obtain that

(3.18) $H_{s}=e^{f_{s}\circ\Phi}l_{F}(\sigma)\ \mathrm{in}\ U_{1},\
H_{s}|_{S^{1}\times\\{0\\}}=l_{F}(\sigma),\ dH_{s}|_{S^{1}\times\\{0\\}}=0.$

Write $\alpha_{s}=H_{s}\alpha_{st}$ with $\alpha_{st}=dz+x_{1}dx_{2}$. The
Reeb vector field of $\alpha_{s}$ is

(3.19) $\displaystyle R_{\alpha_{s}}:$
$\displaystyle=(R_{\alpha_{s},z},R_{\alpha_{s},x_{1}},R_{\alpha_{s},x_{2}})$
$\displaystyle=H_{s}^{-2}(H_{s}+x_{1}H_{s,x_{1}}^{\prime},H_{s,x_{2}}^{\prime}-x_{1}H_{s,z}^{\prime},-H_{s,x_{1}}^{\prime}).$

Indeed,
$\alpha_{s}(R_{\alpha_{s}})=H_{s}^{-1}(H_{s}+x_{1}H_{s,x_{1}}^{\prime}+x_{1}(-H_{s,x_{1}}^{\prime}))=1$
and

$\displaystyle\iota_{R_{\alpha_{s}}}d(\alpha_{s})$
$\displaystyle=(dH_{s}\wedge\alpha_{st}+H_{s}dx_{1}\wedge
dx_{2})(R_{\alpha_{s}},\cdot)$
$\displaystyle=dH_{s}(R_{\alpha_{s}})\alpha_{st}-(R_{\alpha_{s},z}+x_{1}R_{\alpha_{s},x_{2}})dH_{s}+H_{s}(R_{\alpha_{s},x_{1}}dx_{2}-R_{\alpha_{s},x_{2}}dx_{1})$
$\displaystyle=H_{s}^{-1}H_{s,z}^{\prime}\alpha_{st}-H_{s}^{-1}H_{s,z}^{\prime}\alpha_{st}=0.$

The last equality follows from the identities

$\displaystyle dH_{s}(R_{\alpha_{s}})=$ $\displaystyle
H_{s}^{-2}(H_{s,z}^{\prime}(H_{s}+x_{1}H_{s,x_{1}}^{\prime})+H_{s,x_{1}}^{\prime}(H_{s,x_{2}}^{\prime}-x_{1}H_{s,z}^{\prime})+H_{s,x_{2}}^{\prime}(-H_{s,x_{1}}^{\prime}))$
$\displaystyle=$ $\displaystyle H_{s}^{-1}H_{s,z}^{\prime}.$

and

$\displaystyle(R_{\alpha_{s},z}+x_{1}R_{\alpha_{s},x_{2}})dH_{s}-H_{s}(R_{\alpha_{s},x_{1}}dx_{2}-R_{\alpha_{s},x_{2}}dx_{1})$
$\displaystyle=$ $\displaystyle
H_{s}^{-2}(H_{s}dH_{s}-H_{s}(H_{s,x_{2}}^{\prime}-x_{1}H_{s,z}^{\prime})dx_{2}+H_{s}(-H_{s,x_{1}}^{\prime})dx_{1})$
$\displaystyle=$ $\displaystyle H_{s}^{-1}H_{s,z}^{\prime}\alpha_{st},$

where $R_{\alpha_{s},z}+x_{1}R_{\alpha_{s},x_{2}}=H_{s}^{-1}$ by (3.19). Then
write $x_{s}=(x_{s,1},x_{s,2})$, we have

(3.20)
$\dot{x}_{s}(t)=H_{s}^{-2}(H_{s,x_{2}}^{\prime}-x_{1}H_{s,z}^{\prime},-H_{s,x_{1}}^{\prime})(z_{s}(t),x_{s}(t)),\
\forall t\in[0,1].$

By linearizing (3.20) at $S^{1}\times\\{0\\}$ along coordinates
$x=(x_{1},x_{2})$, we obtain that

(3.21) $\displaystyle\dot{R}_{s}(t)$
$\displaystyle=\frac{1}{l_{F}(\sigma)^{2}}\begin{pmatrix}(H_{s})_{x_{1}x_{2}}^{\prime\prime}&(H_{s})_{x_{2}x_{2}}^{\prime\prime}\\\
-(H_{s})_{x_{1}x_{1}}^{\prime\prime}&-(H_{s})_{x_{1}x_{2}}^{\prime\prime}\end{pmatrix}(t,0,0)R_{s}(t)$
$\displaystyle=\frac{1}{l_{F}(\sigma)^{2}}J(-\nabla_{x}^{2}H+\frac{sl_{F}(\sigma)}{2}\nabla^{2}_{x}\tilde{h})(t,0,0)R_{s}(t),\
R_{s}(0)=I_{2},$

where $\tilde{h}=h\circ\Phi$. By using (3.17), the last equality follows from
the identities

$\nabla_{x}^{2}H_{s}(t,0,0)=(\nabla_{x}^{2}H+l_{F}(\sigma)\nabla_{x}^{2}(f_{s}\circ\Phi))(t,0,0),$
$\nabla_{x}^{2}(f_{s}\circ\Phi)(t,0,0)=-\frac{s\nabla_{x}^{2}\tilde{h}}{2(1+s\tilde{h})}(t,0,0),$

and $h|_{\gamma_{\sigma}}=0$ by assumption. By taking the differential of
(3.21) at $s=0$, we have

(3.22) $\frac{\partial\dot{R}_{s}(t)}{\partial
s}|_{s=0}=\frac{1}{l_{F}(\sigma)^{2}}(\frac{l_{F}(\sigma)}{2}J\nabla^{2}_{x}\tilde{h}(t,0,0)R_{0}(t)-J\nabla^{2}_{x}H(t,0,0)\frac{\partial
R_{s}(t)}{\partial s}|_{s=0}).$

It’s worth noting that, by (3.17), we have

(3.23) $\nabla^{2}_{x}H(t,0,0)=0,\ \forall
t\in(t_{0}-\epsilon,t_{0}+\epsilon),\ \nabla^{2}_{x}\tilde{h}(t,0,0)=0,\
\forall t\notin(t_{0}-\epsilon,t_{0}+\epsilon).$

Therefore, we have

(3.24) $R_{0}(t)=R_{0}(t_{0}),\ \forall t\in(t_{0}-\epsilon,t_{0}+\epsilon).$

Besides, for any $s\in[0,1]$, (3.23) also implies that

(3.25) $R_{s}(t)=\left\\{\begin{aligned} &R_{0}(t),\ \forall
t\in[0,t_{0}-\epsilon]\\\
&R_{0}(t)R_{0}(t_{0}+\epsilon)^{-1}R_{s}(t_{0}+\epsilon),\ \forall
t\in[t_{0}+\epsilon,1],\end{aligned}\right.$

By definition (2.1), (3.13) and local model (3.16), we have

(3.26) $P_{\sigma}(f_{s})=R_{s}(1),\forall s\in[0,1],\ \mathrm{especially},\
P_{\sigma}(0)=R_{0}(1).$

After taking the differential of $P_{\sigma}(f_{s})$ at $s=0$ and denoting by

(3.27)
$C:=\frac{1}{2l_{F}(\sigma)}\int^{t_{0}+\epsilon}_{t_{0}-\epsilon}J\nabla^{2}_{x}\tilde{h}(t,0,0)R_{0}(t)dt,$

we can conclude that

(3.28) $\displaystyle\quad\ dP_{\sigma}|_{f=0}(h)=\frac{\partial
R_{s}(1)}{\partial s}|_{s=0}-\frac{\partial R_{s}(0)}{\partial
s}|_{s=0}=\int^{1}_{0}\frac{\partial\dot{R}_{s}(t)}{\partial s}|_{s=0}dt$
$\displaystyle=\frac{1}{l_{F}(\sigma)^{2}}\int^{1}_{0}\frac{l_{F}(\sigma)}{2}J\nabla^{2}_{x}\tilde{h}(t,0,0)R_{0}(t)-J\nabla^{2}_{x}H(t,0,0)\frac{\partial
R_{s}(t)}{\partial s}|_{s=0}dt$
$\displaystyle=C-\int^{1}_{t_{0}+\epsilon}J\nabla^{2}_{x}H(t,0,0)\frac{\partial
R_{s}(t)}{\partial s}|_{s=0}dt,$

in which the first equality follows from $R_{s}(0)\equiv I_{2}$, the third
equality is held by (3.22) and the last equality is obtained from (3.23) and
(3.25).

Furthermore, by using (3.25), (3.21) and (3.23), we have

(3.29) $\displaystyle\quad\
R_{s}(t_{0}+\epsilon)-R_{s}(t_{0}-\epsilon)=\int^{t_{0}+\epsilon}_{t_{0}-\epsilon}\dot{R}_{s}(t)dt$
$\displaystyle=\frac{1}{l_{F}(\sigma)^{2}}\int^{t_{0}+\epsilon}_{t_{0}-\epsilon}J(\frac{sl_{F}(\sigma)}{2}\nabla^{2}_{x}\tilde{h}-\nabla^{2}_{x}H)(t,0,0)R_{s}(t)dt$
$\displaystyle=\int^{t_{0}+\epsilon}_{t_{0}-\epsilon}\frac{s}{2l_{F}(\sigma)}J\nabla^{2}_{x}\tilde{h}(t,0,0)R_{s}(t)dt.$

Then, by taking the differential of (3.29) at $s=0$, (3.27) implies that

(3.30) $\frac{\partial R_{s}(t_{0}+\epsilon)}{\partial
s}|_{s=0}=\frac{1}{2l_{F}(\sigma)}\int^{t_{0}+\epsilon}_{t_{0}-\epsilon}J\nabla^{2}_{x}\tilde{h}(t,0,0)R_{0}(t)dt=C.$

Applying (3.25) and (3.30), we conclude that, for any
$t\in[t_{0}+\epsilon,1]$,

(3.31) $\frac{\partial R_{s}(t)}{\partial
s}|_{s=0}=R_{0}(t)R_{0}(t_{0}+\epsilon)^{-1}\frac{\partial
R_{s}(t_{0}+\epsilon)}{\partial
s}|_{s=0}=R_{0}(t)R_{0}(t_{0}+\epsilon)^{-1}C.$

Finally, we have

$\displaystyle dP_{\sigma}|_{0}(h)$
$\displaystyle=C+\int^{1}_{t_{0}+\epsilon}-J\nabla^{2}_{x}H(t,0,0)R_{0}(t)R_{0}(t_{0}+\epsilon)^{-1}Cdt$
$\displaystyle=C+(R_{0}(1)-R_{0}(t_{0}+\epsilon))R_{0}(t_{0}+\epsilon)^{-1}C=R_{0}(1)R_{0}(t_{0}+\epsilon)^{-1}C$
$\displaystyle=P_{\sigma}(0)R_{0}(t_{0})^{-1}(\int^{t_{0}+\epsilon}_{t_{0}-\epsilon}\frac{J\nabla^{2}\tilde{h}}{2l_{F}(\sigma)}(t,0,0)dt)R_{0}(t_{0}),$

where the first equality follows from (3.28), (3.31), the second equality is
held by (3.21) with $s=0$, the last equality is given by (3.24), (3.26) and
(3.27). The result holds. ∎

Now we prove the following lemma.

######  Lemma 3.2.

Let $F\in\mathcal{F}(\Sigma)$ and
$\sigma_{1},\cdots,\sigma_{k}\in\mathrm{CG}(\Sigma,F)$. Then for any
$\epsilon>0$, there exists a $F^{\prime}\in\mathcal{F}(\Sigma)$ with
$\|F-F^{\prime}\|_{C^{\infty}}<\epsilon$ such that
$\sigma_{1},\cdots,\sigma_{k}\in\mathrm{CG}(\Sigma,F^{\prime})$ are
nondegenerate closed geodesics. Especially, if $F$ is in
$\mathcal{F}^{s}(\Sigma)$ or $\mathcal{M}(\Sigma)$, so is $F^{\prime}$.

###### Proof.

For any $\sigma_{i}$ with length $l_{F}(\sigma_{i})$, we choose a disk-like
open neighborhood $U_{i,1}$ of $\gamma_{\sigma_{i}}(l_{F}(\sigma_{i})t_{i,0})$
in its tubular neighborhood $U_{i}$, which satisfies

(3.32)
$U_{i,1}\cap\gamma_{\sigma_{i}}=\gamma_{\sigma_{i}}(l_{F}(\sigma_{i})(t_{i,0}-\epsilon),l_{F}(\sigma_{i})(t_{i,0}+\epsilon)),\
\forall i=1,\cdots,k,$

with $\pi(U_{i,1})\cap\sigma_{j}=\emptyset$ and
$\pi(U_{i,1})\cap\pi(U_{j,1})=\emptyset$ for any $j\neq i$ and

$(l_{F}(\sigma_{i})(t_{i,0}-\epsilon),l_{F}(\sigma_{i})(t_{i,0}+\epsilon))\subset(0,l_{F}(\sigma_{i}))\cap\mathrm{reg}(\sigma_{i}).$

Let $(q,p)$ be the local coordinates in $U_{i}$. Similar to (3.3), we choose

(3.33)
$f_{s}=-\frac{1}{2}\ln(1+s(\tilde{A}(q)p,p))\in\mathcal{F}^{pert}_{\mathcal{U}_{1}},\
\mathcal{U}_{1}=\cup_{i=1}^{k}\pi(U_{i,1}),$

i.e. $\tilde{A}(\cdot)$ is supported in $\pi(U_{i,1})$s. Moreover, nearby each
$\sigma_{i}$, we choose $\tilde{A}(\cdot)$ as

(3.34)
$\tilde{A}(q(t_{1},t_{2}))=\begin{pmatrix}a_{i,1}(t_{2})&a_{i,2}(t_{2})t_{1}\\\
a_{i,2}(t_{2})t_{1}&a_{i,3}(t_{2})t_{1}^{2}\end{pmatrix},\
a_{i,\\{1,2,3\\}}\in C^{\infty}_{0}([t_{i,0}-\epsilon,t_{i,0}+\epsilon]),$

under the representation (3.7). Since $\pi(U_{i,1})$ avoids each other and
$\sigma_{j\neq i}$, therefore, the matrization (3.34) can be chosen
independently for each $i$. Since $\pi(U_{i,1})$ also avoids the self-
intersections of $\sigma_{i}$, therefore, $\mathrm{supp}f_{s}$ does not effect
$\sigma_{i}(t)$ for any
$t\notin(l(\sigma_{i})(t_{i,0}-\epsilon),l(\sigma_{i})(t_{i,0}+\epsilon))$.
Then by construction (3.34) and condition (3.12), we also have
$f_{s}|_{\gamma_{\sigma_{i}}}=0=df_{s}|_{\gamma_{\sigma_{i}}}$ and
$\gamma_{\sigma_{i}}\in R(S^{*}_{F}\Sigma,e^{f_{s}}\lambda(F))$. Denote by $h$
the initial velocity of $f_{s}$ as (3.8) and write
$\gamma_{\sigma_{i}}(t)=(\sigma_{i}(t),p_{i}(t))$ as (2.10). (3.11) implies
that

(3.35)
$\nabla^{2}h|_{\xi_{F}|_{\gamma_{\sigma_{i}}(t)}}(\sigma_{i}(t),p_{i}(t))=\begin{pmatrix}-a_{i,1}(t)&-a_{i,2}(t)\\\
-a_{i,2}(t)&-a_{i,3}(t)\end{pmatrix},$

under the basis $\xi_{F}|_{\gamma_{\sigma_{i}}}=\\{\bar{V}_{i},\bar{W}_{i}\\}$
as in (3.9). According to the local model (3.14), we can identify
$\gamma_{\sigma_{i}}=S^{1}\times\\{0\\}$ and
$\xi_{F}|_{\gamma_{\sigma_{i}}}=\\{\frac{\partial}{\partial
x_{1}},\frac{\partial}{\partial x_{2}}\\}$. Then we can further rewrite (3.35)
as

(3.36)
$\nabla^{2}h|_{\xi_{F}|_{\gamma_{\sigma_{i}}(t)}}(\sigma_{i}(t),p_{i}(t))=\begin{pmatrix}\tilde{a}_{i,1}(t)&\tilde{a}_{i,2}(t)\\\
\tilde{a}_{i,2}(t)&\tilde{a}_{i,3}(t)\end{pmatrix},$

under the coordinates $\\{x_{1},x_{2}\\}$. Since $a_{i,\\{1,2,3\\}}$ can be
any element in $C^{\infty}_{0}([t_{i,0}-\epsilon,t_{i,0}+\epsilon])$, so is
$\tilde{a}_{i,\\{1,2,3\\}}$. Then by Lemma 3.1, $dP_{\sigma_{i}}|_{f=0}$ is
surjective. Therefore, comparing to (3.5) and using (3.33), we can perturb $F$
to $F^{\prime}$ with

(3.37)
${F^{\prime}}^{*}|_{q}(p)=(F^{*}|_{q}(p)^{2}+(\tilde{A}(q)p,p))^{\frac{1}{2}},\
\forall(q,p)\in T^{*}\mathcal{U}_{1},$

with $a_{i,\\{1,2,3\\}}$ are sufficiently small in $C^{\infty}$-norm such that
$\\{\gamma_{\sigma_{i}}\\}_{i=1}^{k}$ are all nondegenerate in
$\mathrm{CG}(\Sigma,F^{\prime})$. Finally, the perturbation (3.37) implies
that if $F\in\mathcal{F}^{s}(\Sigma)$ or $\mathcal{M}(\Sigma)$, so is
$F^{\prime}$. The Lemma holds. ∎

Now, we denote by $\mathcal{F}^{N}_{C^{r}}(\Sigma,F)$ the collection of
$C^{r}$-smooth $[0,1]^{N}$-families in $\mathcal{F}(\Sigma)$ with $F$ at
$0\in[0,1]^{N}$ for any $N\in\mathbb{N}_{+}$. Similarly, we have
$\mathcal{F}^{s,N}_{C^{r}}(\Sigma,F)$ and $\mathcal{M}^{N}_{C^{r}}(\Sigma,g)$
for symmetric Finsler and Riemannian metrics. For any
$\\{F_{\tau}\\}_{\tau\in[0,1]^{N}}$ in $\mathcal{F}^{N}_{C^{r}}(\Sigma,F)$
with $F_{0}=F$, by Proposition 2.4 (1), we have a corresponding
$[0,1]^{N}$-family of contact forms $e^{f_{\tau}}\lambda(F)$ in
$\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$, where $f_{\tau}(q,p)=-\ln
F^{*}_{\tau}|_{q}(p)$. Write
$\mathcal{F}^{N}(\Sigma,F):=\mathcal{F}^{N}_{C^{\infty}}(\Sigma,F)$. We next
prove the following Lemma by using some ideas of Lemma 3.2 in [17].

######  Lemma 3.3.

For generic $F_{[0,1]^{N}}\in\mathcal{F}^{N}(\Sigma,F)$, we have a full
measure subset

(3.38) $\\{\tau\in[0,1]^{N}|\ F_{\tau}\ \mathrm{is\
bumpy}\\}\subset[0,1]^{N}.$

So is for generic $F_{[0,1]^{N}}\in\mathcal{F}^{s,N}(\Sigma,F)$ (resp.
$g_{[0,1]^{N}}\in\mathcal{M}^{N}(\Sigma,g)$) if $F$ is in
$\mathcal{F}^{s}(\Sigma)$ (resp. $g=F^{2}$ is in $\mathcal{M}(\Sigma)$).

###### Proof.

Firstly, let $E:\mathcal{F}^{N}_{C^{r}}(\Sigma,F)\times
S^{*}_{F}\Sigma\times[0,1]^{N}\times\mathbb{R}_{+}\rightarrow
S^{*}_{F}\Sigma\times S^{*}_{F}\Sigma$, which is sending
$(F_{[0,1]^{N}},(q_{0},p_{0}),\tau,l)$ to
$((q_{0},p_{0}),(q_{\tau}(l),p_{\tau}(l))=\varphi^{l}_{F_{\tau}^{*}}(q_{0},p_{0}))$,
where $\varphi^{t}_{F_{\tau}^{*}}$ denotes the Reeb flow of
$e^{f_{\tau}}\lambda(F)$ by Proposition 2.4 (1) and (3). Similarly, we have
$E^{s}$ and $E^{R}$ by replacing $\mathcal{F}^{N}_{C^{r}}(\Sigma,F)$ to
$\mathcal{F}^{s,N}_{C^{r}}(\Sigma,F)$ and $\mathcal{M}^{N}_{C^{r}}(\Sigma,g)$.
By definition, $E$ (resp. $E^{s}$ and $E^{R}$) is a $C^{r-1}$-map. We denote
by $\Delta_{F}$ the diagonal of $S^{*}_{F}\Sigma\times S^{*}_{F}\Sigma$ and
denote by $\mathrm{CG}^{p}(\Sigma,F)$ the set of closed geodesics on $\Sigma$
with minimal periods. Now we define an open set

(3.39) $\displaystyle\mathcal{O}:=$
$\displaystyle\\{(F_{[0,1]^{N}},(q_{0},p_{0}),\tau,l)\in\mathrm{Dom}E|$
$\displaystyle(q_{\tau}(l_{1}),p_{\tau}(l_{1}))\neq(q_{\tau}(l_{2}),p_{\tau}(l_{2})),\forall
l_{1}\neq l_{2}\ \mathrm{in}\ [0,\frac{l}{2}]\\}.$

(resp. $\mathcal{O}^{s}\subset\mathrm{Dom}E^{s}$ and
$\mathcal{O}^{R}\subset\mathrm{Dom}E^{R}$ open). Then we have

(3.40) $\displaystyle\quad\ E|_{\mathcal{O}}^{-1}(\Delta_{F})$
$\displaystyle=\\{(F_{[0,1]^{N}},(q_{0},p_{0}),\tau,l)\in\mathcal{O}|\
(q_{0},p_{0})=(q_{\tau}(l),p_{\tau}(l))\\}$
$\displaystyle\cong\\{(F_{[0,1]^{N}},\tau,\sigma)|\
F_{[0,1]^{N}}\in\mathcal{F}^{N}_{C^{r}}(\Sigma,F),\tau\in[0,1]^{N},\sigma\in\mathrm{CG}^{p}(\Sigma,F_{\tau})\\}$
$\displaystyle=:\mathrm{CG}^{p}(\mathcal{F}^{N}_{C^{r}}(\Sigma,F)).$

Similarly, we have the following:

$E^{s}|_{\mathcal{O}^{s}}^{-1}(\Delta_{F})=\mathrm{CG}^{p}(\mathcal{F}^{s,N}_{C^{r}}(\Sigma,F)),\
E^{R}|_{\mathcal{O}^{R}}^{-1}(\Delta_{g})=\mathrm{CG}^{p}(\mathcal{M}^{N}_{C^{r}}(\Sigma,g)).$

######  Claim 1.

$E|_{\mathcal{O}}$ (resp. $E^{s}|_{\mathcal{O}^{s}}$ and
$E^{R}|_{\mathcal{O}^{R}}$) transverses to $\Delta_{F}$ (resp. $\Delta_{F}$
and $\Delta_{g}$).

###### Proof.

Indeed, we firstly prove $E|_{\mathcal{O}}$ transverses to $\Delta_{F}$. let
$(x_{0},x_{0}:=(p_{0},q_{0}))$ be in
$\mathrm{Im}E|_{\mathcal{O}}\cap\Delta_{F}$, where
$(q_{0},p_{0})=\varphi^{l}_{F_{\tau}^{*}}(q_{0},p_{0})$. Then
$T_{(x_{0},x_{0})}\Delta_{F}=\\{(v,v)|\ \forall v\in
T_{x_{0}}S^{*}_{F}\Sigma\\}$, and

(3.41) $\displaystyle T_{(x_{0},x_{0})}\mathrm{Im}E|_{\mathcal{O}}$
$\displaystyle=\\{(v,d\varphi^{t}_{F_{\tau}^{*}}|_{x_{0}}(v))|\ \forall v\in
T_{x_{0}}S^{*}_{F}\Sigma\\}+\\{(0,\frac{d}{dt}|_{t=l}\varphi^{t}_{F_{\tau}^{*}}(x_{0}))\\}$
$\displaystyle+\\{(0,\frac{d}{ds}|_{s=0}\varphi^{l}_{{F^{s}_{\tau}}^{*}}(x_{0}))|\
\forall\\{F^{s}_{[0,1]^{N}}\\}_{s\in(-\epsilon,\epsilon)}\subset\mathcal{F}^{N}_{C^{r}}(\Sigma,F)\\}$
$\displaystyle+\\{(0,\frac{d}{d\tau_{i}}|_{\tau_{i}=0}\varphi^{l}_{F_{\tau}^{*}}(x_{0}))|\
\forall i\in\\{1,\cdots,N\\}\\}.$

After identifying $(S^{*}_{F}\Sigma,e^{f_{\tau}}\lambda(F))$ to
$(S^{*}_{F_{\tau}}\Sigma,\lambda(F_{\tau}))$, we denote by $x_{0}(\mathbb{R})$
the Reeb orbit of $(S^{*}_{F_{\tau}}\Sigma,\lambda(F_{\tau}))$ starting from
$x_{0}$ with minimal period $l$ and let $U_{0}$ be an open small tubular
neighborhood of $x_{0}(\mathbb{R})$ with local coordinates
$(q_{\tau},p_{\tau})$. Furthermore, we choose a smaller disk-like open
neighborhood $U_{0,1}\subset U_{0}$, which satisfies

(3.42) $U_{0,1}\cap
x_{0}(\mathbb{R})=x_{0}(lt_{0}-l\epsilon,lt_{0}+l\epsilon),$

with $(lt_{0}-l\epsilon,lt_{0}+l\epsilon)\subset(0,l)\cap\mathrm{reg}(\pi\circ
x_{0}(\mathbb{R}))$ as (3.32).

Now, we let

(3.43) $\lambda^{s}(F_{\tau}):=e^{f_{s}}\lambda(F_{\tau}),\ s\in[0,\epsilon),$

be a path in $\Lambda^{Fin}(S^{*}_{F_{\tau}}\Sigma,\xi_{F_{\tau}})$, where

(3.44)
$f_{s}(q_{\tau},p_{\tau})=-\frac{1}{2}\ln(1+s(\tilde{A}(q_{\tau})p_{\tau},p_{\tau}))\in\mathcal{F}_{\pi(U_{0,1})}^{pert},$

i.e. $\tilde{A}$ is supported $\pi(U_{0,1})$. The initial velocity of $f_{s}$
becomes

(3.45) $h(q_{\tau},p_{\tau})=\frac{\partial f_{s}}{\partial
s}|_{s=0}(q_{\tau},p_{\tau})=-\frac{1}{2}(\tilde{A}(q_{\tau})p_{\tau},p_{\tau}).$

Moreover, near $\pi\circ x_{0}(\mathbb{R})$, we further choose
$\tilde{A}(\cdot)$ as

(3.46)
$\tilde{A}(q_{\tau}(t_{1},t_{2}))=\begin{pmatrix}b_{1}(t_{2})&-b_{3}(t_{2})\\\
-b_{3}(t_{2})&2b_{2}(t_{2})t_{1}\end{pmatrix},\ b_{1,2,3}\in
C^{\infty}_{0}([lt_{0}-l\epsilon,lt_{0}+l\epsilon]),$

under the representation (3.7). Let $\mathcal{R}_{s}$ be the Reeb vector field
of $\lambda^{s}(F_{\tau})$, then

(3.47) $\mathcal{R}_{s}=e^{-f_{s}}(\mathcal{R}_{0}-X_{f_{s}}),\
s\in[0,\epsilon),$

where $X_{f_{s}}\in\xi_{F_{\tau}}$ is uniquely determined by
$\iota_{X_{f_{s}}}d\lambda(F_{\tau})=-df_{s}$ on $\xi_{F_{\tau}}$. Indeed,
directly computation shows that
$\lambda^{s}(F_{\tau})(\mathcal{R}_{s})=\lambda(F_{\tau})(\mathcal{R}_{0}-X_{f_{s}})=1$
and

$\displaystyle\iota_{\mathcal{R}_{s}}d(\lambda^{s}(F_{\tau}))$
$\displaystyle=(df_{s}\wedge\lambda(F_{\tau})+d\lambda(F_{\tau}))(\mathcal{R}_{0}-X_{f_{s}},\cdot)$
$\displaystyle=df_{s}(\mathcal{R}_{0}-X_{f_{s}})\lambda(F_{\tau})-\lambda(F_{\tau})(\mathcal{R}_{0}-X_{f_{s}})df_{s}+d\lambda(F_{\tau})(\mathcal{R}_{0}-X_{f_{s}},\cdot)$
$\displaystyle=-df_{s}+df_{s}=0,$

on $\xi_{F_{\tau}}=\ker\lambda(F_{\tau})$. Let
$x_{s}(t)=\varphi^{t}_{\lambda^{s}(F_{\tau})}(x_{0})$ be the solution of

(3.48)
$\dot{x}_{s}(t)=\mathcal{R}_{s}(x_{s}(t))=e^{-f_{s}(x_{s}(t))}(\mathcal{R}_{0}(x_{s}(t))-X_{f_{s}}(x_{s}(t))),\
x_{s}(0)=x_{0}.$

By taking the differential of (3.48) at $s=0$ and then using $f_{0}=0$,
$X_{f_{0}}=0$ and

(3.49) $\iota_{X_{h}}d\lambda(F_{\tau})=-\frac{\partial df_{s}}{\partial
s}|_{s=0}=-dh,\ X_{h}:=\frac{\partial X_{f_{s}}}{\partial
s}|_{s=0}\in\xi_{F_{\tau}},$

we obtain that

(3.50)
$\dot{\eta}(t)=-h(x_{0}(t))\mathcal{R}_{0}|_{x_{0}(t)}-X_{h}|_{x_{0}(t)}+\nabla\mathcal{R}_{0}(\eta(t)),\
\eta(t)=\frac{\partial x_{s}(t)}{\partial s}|_{s=0}.$

Firstly, by (3.46) and (3.45), for any
$t\in(lt_{0}-l\epsilon,lt_{0}+l\epsilon)$, we have

(3.51)
$h(x_{0}(t))=-\frac{1}{2}(\tilde{A}(q_{\tau}(t))p_{\tau}(t),p_{\tau}(t))=-b_{2}(t)t_{1}|_{t_{1}=0}=0,$

where $x_{0}(t)=(q_{\tau}(t),p_{\tau}(t))$ in $U_{0,1}$. So is for
$t\notin(lt_{0}-l\epsilon,lt_{0}+l\epsilon)$. Besides, under the symplectic
basis $\\{\bar{V}_{\tau},\bar{W}_{\tau}\\}$ of
$(\xi_{F_{\tau}},d\lambda(F_{\tau}))$ as in (3.9), the gradient of $h$ can be
written as

$\displaystyle dh(x_{0}(t))|_{\xi_{F_{\tau}}}=(b_{3}(t),-b_{2}(t)),\ \forall
t\in(lt_{0}-l\epsilon,lt_{0}+l\epsilon),$

as the formula (3.10). Then by (3.49), we have
$X_{h}|_{x_{0}(t)}=(b_{2}(t),b_{3}(t))$. As the local model (3.14), we can
construct a local model of $U_{0}$ by isomorphism

(3.52) $\Phi_{\tau}:(S^{1}\times
D^{2}_{\epsilon},H_{\tau}(dz+x_{1}dx_{2}))\cong(U_{0},\lambda(g_{\tau}))((z,x_{1},x_{2})\mapsto(q_{\tau},p_{\tau})),$

where $x_{0}(lt)=\Phi_{\tau}(t,0,0)$ and (after identifying $U_{0,1}$ with
$\Phi_{\tau}^{-1}(U_{0,1})$)

(3.53) $H_{\tau}|_{U_{0,1}}\equiv l,\ H_{\tau}|_{S^{1}\times\\{0\\}}=l,\
dH_{\tau}|_{S^{1}\times\\{0\\}}=0.$

Under the coordinates $\\{x_{1},x_{2}\\}$, we can rewrite

(3.54) $X_{h}|_{x_{0}(t)}=(\tilde{b}_{2}(t),\tilde{b}_{3}(t)),\
\tilde{b}_{2,3}\in C^{\infty}_{0}([t_{0}-\epsilon,t_{0}+\epsilon]).$

Next, since
$\mathcal{R}_{0}=H_{\tau}^{-2}(H_{\tau}+x_{1}H_{\tau,x_{1}}^{\prime},H_{\tau,x_{2}}^{\prime}-x_{1}H_{\tau,z}^{\prime},-H_{\tau,x_{1}}^{\prime})$
as in (3.19), then

(3.55) $\displaystyle\quad\
\nabla\mathcal{R}_{0}|_{(t,0,0)}(\eta(t))=l^{-2}(0,dH_{\tau,x_{2}}^{\prime}|_{(t,0,0)}\eta(t),-dH_{\tau,x_{1}}^{\prime}|_{(t,0,0)}\eta(t))$
$\displaystyle=l^{-2}(0,H_{\tau,x_{2}x_{1}}^{\prime\prime}\eta_{x_{1}}+H_{\tau,x_{2}x_{2}}^{\prime\prime}\eta_{x_{2}},-H_{\tau,x_{1}x_{1}}^{\prime\prime}\eta_{x_{1}}-H_{\tau,x_{1}x_{2}}^{\prime\prime}\eta_{x_{2}})|_{(t,0,0)}(t)$
$\displaystyle=l^{-2}(0,(-J\nabla^{2}_{x}H_{\tau}(t,0,0)\eta_{x}(t))^{T}),\
\eta_{x}(t)=(\eta_{x_{1}},\eta_{x_{2}})^{T}(t),$

in which, the first equality follows from (3.53), the second equality is
implied by
$H_{\tau,x_{1}z}^{\prime\prime}(t,0,0)=0=H_{\tau,x_{2}z}^{\prime\prime}(t,0,0)$,
which comes from (3.53), either. Then by (3.51), (3.54) and (3.55), (3.50) can
be rewritten as $\dot{\tilde{\eta}}_{z}(t)=0$
($\tilde{\eta}=(\tilde{\eta}_{z},\tilde{\eta}_{x})=d\Phi_{\tau}\circ\eta$) and

$\dot{\tilde{\eta}}_{x}(t)=-\tilde{b}(t)-l^{-2}J\nabla^{2}_{x}H_{\tau}(t,0,0){\tilde{\eta}}_{x}(t),\
\tilde{b}(t)=(\tilde{b}_{2},\tilde{b}_{3})^{T}(t).$

Since $\tilde{\eta}(0)=\frac{\partial\Phi_{\tau}\circ x_{s}(0)}{\partial
s}|_{s=0}=0$ by (3.48), then

(3.56) $\tilde{\eta}_{z}(t)=0,\
\tilde{\eta}_{x}(t)=-R_{\tau}(t)\int^{t}_{0}R_{\tau}(s)^{-1}\tilde{b}(s)ds,$

where $R_{\tau}$ solves
$\dot{R}_{\tau}(t)=-l^{-2}J\nabla^{2}_{x}H_{\tau}(t,0,0)R_{\tau}(t)$ and
$R_{\tau}(0)=I_{2}$. By (3.53), we can further obtain that

$\nabla^{2}_{x}H_{\tau}(t,0,0)=0,\ R_{\tau}(t)=R_{\tau}(t_{0}),$

for any $t\in(t_{0}-\epsilon,t_{0}+\epsilon)$ as (3.24). Then from (3.56), we
have $\tilde{\eta}_{z}(1)=0$ and

(3.57)
$\tilde{\eta}_{x}(1)=-R_{\tau}(1)R_{\tau}(t_{0})^{-1}\int^{t_{0}+\epsilon}_{t_{0}-\epsilon}\tilde{b}(s)ds.$

Since $\tilde{b}_{2,3}$ can be any element in
$C^{\infty}_{0}([t_{0}-\epsilon,t_{0}+\epsilon])$, then similar to (3.5), we
let

(3.58)
${F^{s}_{\tau}}^{*}|_{q_{\tau}}(p_{\tau})=(F_{\tau}|_{q_{\tau}}(p_{\tau})^{2}+s(\tilde{A}(q_{\tau})p_{\tau},p_{\tau}))^{\frac{1}{2}},\
\forall(q_{\tau},p_{\tau})\in T^{*}\pi(U_{0,1}),$

with $\tilde{A}(\cdot)$ as (3.46) and ${F^{s}_{\tau}}^{*}=F^{*}_{\tau}$
elsewhere, we have

(3.59)
$\frac{d}{ds}|_{s=0}\varphi^{l}_{{F^{s}_{\tau}}^{*}}(x_{0})=\frac{d}{ds}|_{s=0}\varphi^{l}_{\lambda^{s}(F_{\tau})}(x_{0})=\eta(l)=d\Phi_{\tau}^{-1}(\tilde{\eta}_{z}(1),\tilde{\eta}_{x}(1)),$

which can be any element in $\xi_{F_{\tau}}|_{x_{0}}$ by (3.57), so is in
$\xi_{F}|_{x_{0}}$ under the identification between
$(S^{*}_{F}\Sigma,e^{f_{\tau}}\lambda(F))$ and
$(S^{*}_{F_{\tau}}\Sigma,\lambda(F_{\tau}))$. Note that the first equality of
(3.59) follows form Proposition 2.4 (1), (3) and (3.43) and the rests follow
from definition of $x_{s},\ \eta$ and local model (3.52). Therefore, by
(3.41), we have

$T_{(x_{0},x_{0})}\mathrm{Im}E|_{\mathcal{O}}+T_{(x_{0},x_{0})}\Delta_{F}=T_{x_{0}}S^{*}_{F}\Sigma\times
T_{x_{0}}S^{*}_{F}\Sigma.$

Especially, since the variation $F_{\tau}^{s}$ defined by (3.58) is also in
$\mathcal{F}^{s}(\Sigma)$ (resp. $\mathcal{M}(\Sigma)$), whenever $F_{\tau}$
is in $\mathcal{F}^{s}(\Sigma)$ (resp. $g_{\tau}=F_{\tau}^{2}$ is in
$\mathcal{M}(\Sigma)$), then after replacing Proposition 2.4 to Proposition
2.5, (3.59) also implies the transversality between $E^{s}|_{\mathcal{O}^{s}}$
(resp. $E^{R}|_{\mathcal{O}^{R}}$) and $\Delta_{F}$ (resp. $\Delta_{g}$).
Claim 1 is true. ∎

Then by Lemma 5.4 and (3.40), Claim 1 implies that
$CG^{p}(\mathcal{F}^{N}_{C^{r}}(\Sigma,F))$ is a $C^{r-1}$-submanifold of
$\mathrm{Dom}E$ with codimension 3. Since the projection
$\mathrm{Dom}E\rightarrow\mathcal{F}^{N}_{C^{r}}(\Sigma,F)$ is Fredholm with
index $N+4$, then the 3-codimensional restriction

(3.60)
$P_{1}:\mathrm{CG}^{p}(\mathcal{F}^{N}_{C^{r}}(\Sigma,F))\rightarrow\mathcal{F}^{N}_{C^{r}}(\Sigma,F)((F_{[0,1]^{N}},\tau,\sigma)\mapsto
F_{[0,1]^{N}})$

is a $C^{r-1}$-Fredholm map with $\mathrm{Ind}P_{1}=N+1$.

Secondly, we consider a $\mathrm{Sp}(2)$-bundle
$\mathcal{E}\rightarrow\mathrm{CG}^{p}(\mathcal{F}^{N}_{C^{r}}(\Sigma,F))$
with fiber

$\mathcal{E}_{(F_{[0,1]^{N}},\tau,\sigma)}=\mathrm{Sp}(\xi_{\gamma_{\sigma}(0)},d\lambda(F_{\tau}))\cong\mathrm{Sp}(2).$

We also denote by $\mathcal{E}^{s},\mathcal{E}^{R}$ the restriction of
$\mathcal{E}$ on $\mathrm{CG}^{p}(\mathcal{F}^{s,N}_{C^{r}}(\Sigma,F))$,
$\mathrm{CG}^{p}(\mathcal{M}^{N}_{C^{r}}(\Sigma,g))$. By theorem 1.7.3 of
[22], any element $M\in\mathrm{Sp}(2)$ is symplectically similar to one of the
following matrices

$N(\pm 1,b)=\begin{pmatrix}\pm 1&b\\\ 0&\pm 1\end{pmatrix},\
D(a)=\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix},\
R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix},$

for any $b\in\\{0,\pm 1\\},a\in\mathbb{R}\setminus\\{0,\pm 1\\}$ and
$\theta\in(0,\pi)\cup(\pi,2\pi)$. Denote by $\mathrm{Sp}_{\pm 1,b}(2)$ (resp.
$\mathrm{Sp}_{a}(2)$ and $\mathrm{Sp}_{\theta}(2)$) the collection of
$M\in\mathrm{Sp}(2)$ with normal form $N(\pm 1,b)$ (resp. $D(a)$ and
$R(\theta)$). We note that $\mathrm{Sp}_{\pm 1,0}(2)$ are 0-dimensional,
$\mathrm{Sp}_{\pm 1,b\neq 0}(2)$, $\mathrm{Sp}_{a}(2)$ and
$\mathrm{Sp}_{\theta}(2)$ are 2-dimensional. Then we have subbundles
$\mathcal{E}_{\pm 1,b}\subset\mathcal{E}$ (resp. $\mathcal{E}_{a}$ and
$\mathcal{E}_{\theta}$) with fiber as $\mathrm{Sp}_{\pm 1,b}(2)$ (resp.
$\mathrm{Sp}_{a}(2)$ and $\mathrm{Sp}_{\theta}(2)$). In addition, there is a
$C^{r-2}$-section of $\mathcal{E}$ induced by the linear Poincaré maps (see
(2.11))

$s:\mathrm{CG}^{p}(\mathcal{F}^{N}_{C^{r}}(\Sigma,F))\rightarrow\mathcal{E},\
(F_{[0,1]^{N}},\tau,\sigma)\mapsto
P_{\sigma}\in\mathrm{Sp}(\xi_{F_{\tau}}|_{\gamma_{\sigma}(0)},d\lambda(F_{\tau})).$

By projecting the intersection $\mathrm{Im}s\cap\mathcal{E}_{*}$, we define

$\mathrm{CG}^{p}_{*}(\mathcal{F}^{N}_{C^{r}}(\Sigma,F)):=\\{(F_{[0,1]^{N}},\tau,\sigma)\in\mathrm{CG}^{p}(\mathcal{F}^{N}_{C^{r}}(\Sigma,F))|\
P_{\sigma}\in\mathrm{Sp}_{*}(2)\\},$

for $*=\\{\pm 1,b\\},a,\theta$, respectively.

Similarly, we have $\mathcal{E}^{s}_{*}$, $\mathcal{E}^{R}_{*}$ and
$\mathrm{CG}^{p}_{*}(\mathcal{F}^{s,N}_{C^{r}}(\Sigma,F))$,
$\mathrm{CG}^{p}_{*}(\mathcal{M}^{N}_{C^{r}}(\Sigma,g))$. We also denote by
$s^{s}$ (resp. $s^{R}$) the restriction of $s$ on subbundle $\mathcal{E}^{s}$
(resp. $\mathcal{E}^{R}$).

######  Claim 2.

Section $s$ transverses to $\mathcal{E}_{*}$ (resp. $\mathcal{E}^{s}_{*}$ and
$\mathcal{E}^{R}_{*}$) in $\mathcal{E}$ (resp. $\mathcal{E}^{s}$ and
$\mathcal{E}^{R}$), for any $*=\\{\pm 1,b\\},a,\theta$, and any $b\in\\{0,\pm
1\\},a\in\mathbb{R}\setminus\\{0,\pm 1\\}$ and
$\theta\in(0,\pi)\cup(\pi,2\pi)$.

###### Proof.

Indeed, without loss of generality, let $(F_{[0,1]^{N}},\tau,\sigma)$ be
arbitrary element in $\mathrm{Im}s\cap\mathcal{E}_{\theta}$ with
$P_{\sigma}\in\mathrm{Sp}_{\theta}(2)$. Similar to Lemma 3.2, we can perturb
$F_{\tau}$ to $F^{s}_{\tau}$ with ${F^{s}_{\tau}}^{*}$ as (3.58), in which
$\tilde{A}(\cdot)$ is supported in a smaller disk-like open neighborhood
$\pi(U_{1})$ of $\sigma(l_{F}(\sigma)t_{0})$ with satisfying (3.15) and has
the form

(3.61)
$\tilde{A}(q(t_{1},t_{2}))=\begin{pmatrix}a_{1}(t_{2})&a_{2}(t_{2})t_{1}\\\
a_{2}(t_{2})t_{1}&a_{3}(t_{2})t_{1}^{2}\end{pmatrix},\ a_{1,2,3}\in
C^{\infty}_{0}([t_{0}-\epsilon,t_{0}+\epsilon]),$

nearby $\sigma$ as in (3.34) under the representation (3.7). Then Lemma 3.1
and the flexibility of $a_{1,2,3}$ imply that the linear Poincaré map
$P_{\sigma}(\cdot)$ of (3.13) is surjective at $0$, which means that, one can
perturb the family $F_{[0,1]^{N}}$ to $F_{[0,1]^{N}}^{s}$ inside of
$\mathcal{F}^{N}_{C^{r}}(\Sigma,F)$ nearby $F_{\tau}$ such that
$\sigma\in\mathrm{CG}(\Sigma,F^{s}_{\tau})$ and $P_{\sigma}(f_{s})$ varies
from $P_{\sigma}(0)$ in arbitrary direction, where $f_{s}$ has the form
(3.44). Therefore, by definition of $\mathcal{E}_{\theta}$, we have

$T_{P_{\sigma}}\mathrm{Im}s+T_{P_{\sigma}}\mathcal{E}_{\theta}=T_{P_{\sigma}}\mathcal{E},$

then $s$ transverses to $\mathcal{E}_{\theta}$ for any
$\theta\in(0,\pi)\cup(\pi,2\pi)$. Similarly, $s$ also transverses to
$\mathcal{E}_{\pm 1,b}$ and $\mathcal{E}_{a}$ for any $b\in\\{0,\pm 1\\}$ and
$a\in\mathbb{R}\setminus\\{0,\pm 1\\}$.

Especially, whenever $F_{\tau}$ is in $\mathcal{F}^{s}(\Sigma)$ (resp.
$g_{\tau}=F_{\tau}^{2}$ is in $\mathcal{M}(\Sigma)$), so is $F^{s}_{\tau}$
(resp. $g_{\tau}^{s}=(F^{s}_{\tau})^{2}$). Therefore, the surjectivity of
(3.13) is always true, so is the transversality between $s^{s}$ (resp.
$s^{R}$) and $\mathcal{E}^{s}_{*}$ (resp. $\mathcal{E}^{R}_{*}$). The Claim 2
holds. ∎

Then by Lemma 5.4, Claim 2 implies that
$\mathrm{CG}^{p}_{*}(\mathcal{F}^{N}_{C^{r}}(\Sigma,F)):=s^{-1}(\mathcal{E}_{*})$
is a $C^{r-2}$-submanifold of
$\mathrm{CG}^{p}(\mathcal{F}^{N}_{C^{r}}(\Sigma,F))$. If $*=\\{\pm 1,b\neq
0\\}$, then

$P_{1,\pm 1,b}:\mathrm{CG}^{p}_{\pm
1,b}(\mathcal{F}^{N}_{C^{r}}(\Sigma,F))\rightarrow\mathcal{F}^{N}_{C^{r}}(\Sigma,F)((F_{[0,1]^{N}},\tau,\sigma)\mapsto
F_{[0,1]^{N}})$

is a 1-codimensional restriction of $P_{1}$ defined in (3.60), then $P_{1,\pm
1,b}$ is a $C^{r-2}$ Fredholm map with $\mathrm{Ind}P_{1,\pm 1,b}=N$.
Similarly, for $*=\\{\pm 1,0\\}$ (resp. $a$ and $\theta$), we have projection
$P_{1,\pm 1,0}$ (resp. $P_{1,a}$ and $P_{1,\theta}$), which is a $C^{r-2}$
Fredholm map with $\mathrm{Ind}P_{1,\pm 1,0}=N-2$ (resp.
$\mathrm{Ind}P_{1,a}=N$ and $\mathrm{Ind}P_{1,\theta}=N$ for any
$a\in\mathbb{R}\setminus\\{0,\pm 1\\}$ and $\theta\in(0,\pi)\cup(\pi,2\pi)$).
If $r>N+2$, then Lemma 5.2 implies that the regular values of $P_{1,\pm 1,b}$,
$P_{1,a}$ and $P_{1,\theta}$ are of second category. Therefore, the subset

(3.62) $\mathcal{F}^{N}_{C^{r},\mathbf{b}}(\Sigma,F):=\bigcap_{b\in\\{0,\pm
1\\}}\mathrm{reg}P_{1,\pm
1,b}\cap\bigcap_{\theta\in\mathbb{Q}\pi}\mathrm{reg}P_{1,\theta}\subset\mathcal{F}^{N}_{C^{r}}(\Sigma,F)$

is of second category, either. In particular,

$\mathrm{CG}^{p}_{*}(\mathcal{F}^{s,N}_{C^{r}}(\Sigma,F)):=s^{-1}(\mathcal{E}^{s}_{*})\
\mathrm{(resp.}\
\mathrm{CG}^{p}_{*}(\mathcal{M}^{N}_{C^{r}}(\Sigma,g)):=s^{-1}(\mathcal{E}^{R}_{*}))$

is also a $C^{r-2}$-submanifold of
$\mathrm{CG}^{p}(\mathcal{F}^{s,N}_{C^{r}}(\Sigma,F))$ (resp.
$\mathrm{CG}^{p}(\mathcal{M}^{N}_{C^{r}}(\Sigma,g))$) and the restricted
projection

$P^{s}_{1,*}:=P_{1,*}:\mathrm{CG}^{p}_{*}(\mathcal{F}^{s,N}_{C^{r}}(\Sigma,F))\rightarrow\mathcal{F}^{s,N}_{C^{r}}(\Sigma,F)\
\mathrm{(resp.}\
P^{R}_{1,*}:=P_{1,*}|_{\mathrm{CG}^{p}_{*}(\mathcal{M}^{N}_{C^{r}}(\Sigma,g))})$

is $C^{r-2}$ Fredholm with index $N-2$, for $*=\\{\pm 1,0\\}$ and $N$
otherwise. If $r>N+2$, the regular values of $P^{s}_{1,*}$ (resp.
$P^{R}_{1,*}$) are of second category. Similar to (3.62), we can also define

$\mathcal{F}^{s,N}_{C^{r},\mathbf{b}}(\Sigma,F)\subset\mathcal{F}^{s,N}_{C^{r}}(\Sigma,F)\
\mathrm{(resp.}\
\mathcal{M}^{N}_{C^{r},\mathbf{b}}(\Sigma,g)\subset\mathcal{M}^{N}_{C^{r}}(\Sigma,g))$

which is of second category.

######  Claim 3.

For any $F_{[0,1]^{N}}\in\mathcal{F}^{N}_{C^{r},\mathbf{b}}(\Sigma,F)$ with
$r$ large enough, we have

$\\{\tau\in[0,1]^{N}|\ F_{\tau}\ \mathrm{is\ bumpy}\\}\ \mathrm{has\ full\
measure\ in}\ [0,1]^{N}.$

So is for $\mathcal{F}^{s,N}_{C^{r},\mathbf{b}}(\Sigma,F)$ and
$\mathcal{M}^{N}_{C^{r},\mathbf{b}}(\Sigma,g)$.

###### Proof.

Let $F_{[0,1]^{N}}\in\mathcal{F}^{N}_{C^{r},\mathbf{b}}(\Sigma,F)$, then Lemma
5.3 implies that

$\mathrm{CG}^{p}_{*}(F_{[0,1]^{N}}):=P_{1,*}^{-1}(F_{[0,1]^{N}}),\ *=(\pm
1,b)|_{b\neq 0},(\pm 1,0),\theta\in\mathbb{Q}\pi,$

are $C^{r-2}$ submanifolds of
$\mathrm{CG}^{p}_{*}(\mathcal{F}^{N}_{C^{r}}(\Sigma,F))$ with dimension
$N,N-2,N$, respectively.

Consider the restricted projection

$P_{2,*}(F_{[0,1]^{N}}):\mathrm{CG}^{p}_{*}(F_{[0,1]^{N}})\rightarrow[0,1]^{N}((F_{[0,1]^{N}},\tau,\sigma)\mapsto\tau).$

Since there is a natural time-shifting $S^{1}$-action

$c:\mathrm{CG}^{p}_{*}(F_{[0,1]^{N}})\rightarrow\mathrm{CG}^{p}_{*}(F_{[0,1]^{N}})((F_{[0,1]^{N}},\tau,\sigma)\mapsto(F_{[0,1]^{N}},\tau,c\cdot\sigma),$

which is invariant under the projection $P_{2,*}(F_{[0,1]^{N}})$ for any
$*\in\\{(\pm 1,b)|_{b\in\\{0,\pm 1\\}},\theta\in\mathbb{Q}\pi\\}$. Then
$P_{2,*}(F_{[0,1]^{N}})$ is not immersed anywhere. When $r>2$, then Lemma 5.1
implies that the regular values of $P_{2,*}(F_{[0,1]^{N}})$ has full measure.
Then for any $\tau$ in $\mathrm{reg}P_{2,*}(F_{[0,1]^{N}})$, we have
$\tau\notin\mathrm{Im}P_{2,*}(F_{[0,1]^{N}})$. Therefore, if

$\tau\in\bigcap_{*=\\{\pm 1,b\\}|_{b\in\\{0,\pm
1\\}},\mathbb{Q}\pi}\mathrm{reg}P_{2,*}(F_{[0,1]^{N}}),$

which is also of second category in $[0,1]^{N}$, then $F_{\tau}$ is bumpy,
i.e.

$\mathrm{CG}^{p}(\Sigma,F_{\tau})\cap\mathrm{CG}^{p}_{*}(F_{[0,1]^{N}})=\emptyset,\
*\in\\{(\pm 1,b)|_{b\in\\{0,\pm 1\\}},\theta\in\mathbb{Q}\pi\\}.$

The cases of $\mathcal{F}^{s,N}_{C^{r},\mathbf{b}}(\Sigma,F)$ and
$\mathcal{M}^{N}_{C^{r},\mathbf{b}}(\Sigma,g)$ are similar. Hence, Claim 3
holds. ∎

To extend Claim 3 to $C^{\infty}$, we only need to take the following
intersection

(3.63) $\mathcal{F}^{N}_{\mathbf{b}}(\Sigma,F):=\bigcap_{r\geq
1}\mathcal{F}^{N}_{C^{r},\mathbf{b}}(\Sigma,F),$

which is of second category, either.

To finish the proof of Lemma 3.3. Let $\mu$ be the Lebesgue measure in
$[0,1]^{N}$. For any $L>>1$, we define

$\displaystyle\mathcal{F}^{N}_{L}(\Sigma,F):$
$\displaystyle=\\{F_{[0,1]^{N}}\in\mathcal{F}^{N}(\Sigma,F)|\ \mu\\{\tau|\
\sigma\in\mathrm{CG}^{p}(\Sigma,F_{\tau})\ \mathrm{is\ nondegenerate}$
$\displaystyle\mathrm{whenever}\ l(\sigma)\leq L\\}>1-L^{-1}\\},$

where $\mathcal{F}^{N}(\Sigma,F)=\mathcal{F}^{N}_{C^{\infty}}(\Sigma,F)$ is
the collection of $C^{\infty}$-smooth $[0,1]^{N}$-families in
$\mathcal{F}(\Sigma)$. To show $\mathcal{F}^{N}_{L}(\Sigma,F)$ is open and
dense. By definition (3.63),
$\mathcal{F}^{N}_{\mathbf{b}}(\Sigma,F)\subset\mathcal{F}^{N}_{L}(\Sigma,F)$,
then the denseness holds. To the openness. By definition, it is sufficient to
show that for any $F_{[0,1]^{N}}\in\mathcal{F}^{N}_{L}(\Sigma,F)$, the
nondegeneracy of the closed geodesics in $\mathrm{CG}^{p}(\Sigma,F_{\tau})$
with length $\leq L$ are preserved under small perturbations of
$F_{[0,1]^{N}}$. Indeed, since those
$\sigma\in\mathrm{CG}^{p}(\Sigma,F_{\tau})$ with $l_{F}(\sigma)\leq l$ are
nondegenerate critical points of length function
$l_{F_{\tau}}:\Lambda(\Sigma)\rightarrow\mathbb{R}$ in level set
$l_{F_{\tau}}^{-1}(0,L]$, where $\Lambda(\Sigma)$ denotes the $C^{1}$ loop
space in $\Sigma$, then these closed geodesics must be finitely many and
isolated up to the canonical $S^{1}$-action. Otherwise, there is a sequence
$\sigma_{i}\in\mathrm{CG}^{P}(\Sigma,F_{\tau})$ converging to $\sigma\in
l_{F_{\tau}}^{-1}(0,L]$, which contradicts the nondegeneracy of $\sigma$.
After an arbitrary small perturbation of $F_{\tau}$, $l_{F_{\tau}}(\cdot)$
varies smoothly, then there appears no degenerate critical point in
$l_{F_{\tau}}^{-1}(0,L]$. Therefore, openness follows.

Finally, for any $F_{[0,1]^{N}}\in\cap_{L\geq
1}\mathcal{F}^{N}_{L}(\Sigma,F)$, which is also of second category in
$\mathcal{F}^{N}(\Sigma,F)$, the subset $\\{\tau\in[0,1]^{N}|\ F_{\tau}\
\mathrm{is\ bumpy}\\}$ has full measure in $[0,1]^{N}$, then the proof is
done. In particular, the above argument and Claim 3 are also available for
$\mathcal{F}^{s,N}(\Sigma,F)$ and $\mathcal{M}^{N}(\Sigma,g)$. ∎

##  4\. Proof of main theorems

In this section, we are aiming to prove the main results, Theorem 1.1 and 1.3
for (symmetric) Finsler metrics, by using properties of ECH spectral
invariants in Subsection 2.1, generic results in Section 3 and relationships
between Reeb flow and geodesic flow in Subsection 2.2. Similarly, we can also
prove Theorem 1.2 and 1.4 for Riemannian metrics. In conclusion, we prove that
there generically exists a sequence of equidistributed closed geodesics on
$\Sigma$ with Finsler metric (resp. symmetric Finsler metric and Riemannian
metric).

###### The proof of Theorem 1.1.

Take a sequence of $\\{\psi_{i}\\}_{i\geq 1}$ in
$C^{\infty}(\Sigma,\mathbb{R})\subset C^{\infty}(S^{*}_{F}\Sigma,\mathbb{R})$,
which is $C^{0}$-dense in $C^{\infty}(\Sigma,\mathbb{R})$.

######  Claim 4.

For any $N\in\mathbb{N}_{+}$, $\epsilon>0$ and any open subset
$\mathcal{U}\subset\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$ (resp.
$\Lambda^{sFin}(S^{*}_{F}\Sigma,\xi_{F})$), there exists a contact form
$\lambda\in\mathcal{U}$ and a nondegenerate Reeb 1-current $C\in
L(S^{*}_{F}\Sigma,\lambda)$ such that

$|C(\psi_{i}\lambda)-\int_{S^{*}_{F}\Sigma}\psi_{i}\lambda\wedge
d\lambda|<\epsilon,\ \forall i=1,\cdots,N.$

###### Proof.

Firstly, we assume $\mathcal{U}\subset\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$.
Let $\lambda_{0}\in\mathcal{U}$, then we can identify
$(S^{*}_{F}\Sigma,\lambda_{0})$ to $(S^{*}_{F_{0}}\Sigma,\lambda(F_{0}))$ for
some $F_{0}\in\mathcal{F}(\Sigma)$ and $(S^{*}_{F}\Sigma,\xi_{F})$ becomes
$(S^{*}_{F_{0}}\Sigma,\xi_{F_{0}})$. Since this identification fixes the zero
section $\Sigma$, therefore, $\\{\psi_{i}\\}_{i\geq 1}$ are still
$C^{0}$-dense in $C^{\infty}(\Sigma,\mathbb{R})\subset
C^{\infty}(S^{*}_{F_{0}}\Sigma,\mathbb{R})$. Let
$c_{\varsigma_{k}}(S^{*}_{F_{0}}\Sigma,\lambda(F_{0}))$ be the spectral
invariant defined as in (2.2), where $\\{\varsigma_{k}\\}_{k\geq 1}$ is
obtained from Proposition 2.1 (5) and we also denote by

(4.1)
$\gamma_{k}(\lambda_{0})=\frac{c_{\varsigma_{k}}(S^{*}_{F_{0}}\Sigma,\lambda(F_{0}))}{\sqrt{2k}}-\sqrt{\mathrm{vol}(S^{*}_{F_{0}}\Sigma,\lambda(F_{0}))},\
k\geq 1.$

as in Proposition 2.2. Choose a smooth $[0,1]^{N}$-family of contact forms

(4.2) $\lambda_{\tau}=e^{f_{\tau}}\lambda(F_{0}),\
f_{\tau}=\frac{1}{\hat{c}}\sum_{i=1}^{N}\tau_{i}\psi_{i}\in
C^{\infty}(\Sigma,\mathbb{R}),\ \forall\tau\in[0,1]^{N},$

and let $F_{\tau\in[0,1]^{N}}\in\mathcal{F}^{N}(\Sigma,F_{0})$ be defined as

$F_{\tau}|_{x}(\cdot):=\exp(\frac{1}{\hat{c}}\sum_{i=1}^{N}\tau_{i}\psi_{i}(x))F_{0}|_{x}(\cdot),\
\forall x\in\Sigma.$

By Proposition 2.4 (2), we have
$(S^{*}_{F_{0}}\Sigma,\lambda_{\tau})\cong(S^{*}_{F_{\tau}}\Sigma,\lambda(F_{\tau}))$.
After choosing $\hat{c}$ sufficiently large, we have
$\lambda_{[0,1]^{N}}\subset\mathcal{U}$. Then by Lemma 3.3, there is a smooth
$[0,1]^{N}$-family of contact forms
$\lambda^{\prime}_{\tau\in[0,1]^{N}}={F_{\tau}^{\prime}}^{*}(\cdot)^{-1}\lambda(F_{0})$
nearby $\lambda_{[0,1]^{N}}$ in $\mathcal{U}$, in which
$F_{[0,1]^{N}}^{\prime}\in\mathcal{F}^{N}(\Sigma,F_{0})$ is a perturbation of
$F_{[0,1]^{N}}$, such that

  * (a)

$\mu\\{\tau\in[0,1]^{N}|\ \lambda_{\tau}^{\prime}\ \mathrm{is\
nondegenerate}\\}=1$,

  * (b)

$\|\hat{c}\frac{\partial\lambda_{\tau}^{\prime}}{\partial\tau_{i}}/\lambda_{\tau}^{\prime}-\psi_{i}\|_{C^{0}(S^{*}_{F_{0}}\Sigma,\mathbb{R})}<\check{c},$
$\check{c}$ can be arbitrary small,

where (b) follows from the identity
$\frac{\partial\lambda_{\tau}}{\partial\tau_{i}}=\frac{\partial
f_{\tau}}{\partial\tau_{i}}\lambda_{\tau}=\frac{\psi_{i}}{\hat{c}}\lambda_{\tau}$
and the fact that $\lambda_{\tau},\lambda_{\tau}^{\prime}$ are
$C^{\infty}$-close to each other in
$\Lambda^{Fin}(S^{*}_{F_{0}},\xi_{F_{0}})$, uniformly.

Since $f\mapsto\gamma_{k}(e^{f}\lambda_{0})$ is Lipschitz in
$C^{0}(S^{*}_{F_{0}}\Sigma,[-1,1])$ by Proposition 2.2, and after writing the
perturbed contact forms $\lambda_{\tau}^{\prime}$ of
$(S^{*}_{F_{0}}\Sigma,\xi_{F_{0}})$ as $e^{f_{\tau}^{\prime}}\lambda(F_{0})$,
we have

(4.3)
$|\gamma_{k}(\lambda_{\tau_{1}}^{\prime})-\gamma_{k}(\lambda_{\tau_{2}}^{\prime})|\leq
c(\lambda(F_{0}))\|f_{\tau_{1}}^{\prime}-f_{\tau_{2}}^{\prime}\|\leq
c(\lambda(F_{0}))\max_{i,\tau}\|\frac{\partial
f_{\tau}^{\prime}}{\partial\tau_{i}}\|_{C^{0}}|\tau_{1}-\tau_{2}|.$

Then the composition
$\gamma^{\prime}_{k}:[0,1]^{N}\rightarrow\mathbb{R}(\tau\mapsto\gamma_{k}(\lambda_{\tau}^{\prime}))$
is uniformly lipschitz. By Proposition 2.1 (5), we also have
$\displaystyle\lim_{k\rightarrow+\infty}\|\gamma_{k}^{\prime}\|_{C^{0}}=0$.

As in (3.38), we denote by $\mathcal{S}^{\prime}:=\\{\tau\in[0,1]^{N}|\
\lambda_{\tau}^{\prime}\ \mathrm{is\ nondegenerate}\\}$, which has full
measure in $[0,1]^{N}$. Then we fix a sufficiently large $k$ such that
$\max\gamma_{k}^{\prime}-\min\gamma_{k}^{\prime}$ is arbitrary small.
According to (4.3), Lemma 5.5 and condition (a) above, there exists $N+1$ many
sequences $\\{\tau^{j,m}\\}_{m\geq 1}$ in $\mathcal{S}^{\prime}$,
$j=1,\cdots,N+1$, such that

  * (1)

$\\{\tau^{j,m}\\}_{m\geq 1}$ converges to a point $\tau^{\infty}\in(0,1)^{N}$
for any $j=1,\cdots,N+1$.

  * (2)

$\gamma_{k}^{\prime}$ is differentiable at $\tau^{j,m}$ for any $m\geq 1$ and
$j=1,\cdots,N+1$.

  * (3)

$\\{\nabla\gamma_{k}^{\prime}(\tau^{j,m})\\}_{m\geq 1}$ converges to some
$v_{j}\in\mathbb{R}^{N}$ for any $j$, which satisfies

$d_{\mathbb{R}^{N}}(0,\mathrm{conv}(v_{1},\cdots,v_{N+1}))<\check{c}.$

Denote by
$\mathrm{vol}_{0}(\lambda)=\mathrm{vol}(S^{*}_{F_{0}}\Sigma,\lambda)$.
According to (3) and (4.1), there are a tuple of coefficients
$\\{a_{j}\\}_{j=1}^{N+1}$ with $\sum_{j=1}^{N+1}a_{j}=1$ such that for any
$i=1,\cdots,N$ and $m$ large enough, we have

(4.4) $\displaystyle\check{c}$
$\displaystyle>\sum_{j=1}^{N+1}a_{j}\frac{\partial\gamma^{\prime}_{k}(\tau^{j,m})}{\partial\tau_{i}}=\sum_{j=1}^{N+1}a_{j}(\frac{\partial
c_{\varsigma_{k}}(S^{*}_{F_{0}}\Sigma,\lambda_{\tau^{j,m}}^{\prime})}{\sqrt{2k}\partial\tau_{i}}-\frac{\int_{S^{*}_{F_{0}}\Sigma}\frac{\partial\lambda_{\tau^{j,m}}^{\prime}}{\partial\tau_{i}}\wedge
d\lambda_{\tau^{j,m}}^{\prime}}{\sqrt{\mathrm{vol}_{0}(\lambda_{\tau^{j,m}}^{\prime})}})$
$\displaystyle\geq\sum_{j=1}^{N+1}a_{j}(\frac{C_{j,m}(\frac{\psi_{i}}{\hat{c}}\lambda_{\tau^{j,m}}^{\prime})}{\sqrt{2k}}-\frac{\int_{S^{*}_{F_{0}}\Sigma}\frac{\psi_{i}}{\hat{c}}\lambda_{\tau^{j,m}}^{\prime}\wedge
d\lambda_{\tau^{j,m}}^{\prime}}{\sqrt{\mathrm{vol}_{0}(\lambda_{\tau^{j,m}}^{\prime})}})-\frac{D\check{c}}{\hat{c}},$

where $C_{j,m}\in L(S^{*}_{F_{0}}\Sigma,\lambda_{\tau^{j,m}}^{\prime})$ with
$C_{j,m}(\lambda_{\tau^{j,m}}^{\prime})=c_{\varsigma_{k}}(S^{*}_{F_{0}}\Sigma,\lambda_{\tau^{j,m}}^{\prime})$
is obtained by Proposition 2.3 (1) and

$D=\max_{j,m,k}\\{\frac{c_{\varsigma_{k}}(\lambda_{\tau^{j,m}}^{\prime})}{\sqrt{2k}}+\sqrt{\mathrm{vol}_{0}(\lambda_{\tau^{j,m}}^{\prime})}\\}.$

The last inequality of (4.4) follows from the above conditions (b) and (2) and
Proposition 2.3. By Proposition 2.1 (5) and condition (1) above, the maximum
$D<+\infty$ exists and only depends on $\lambda_{[0,1]^{N}}^{\prime}$.
Therefore, by (4.4), we have

(4.5)
$|\sum_{j=1}^{{N+1}}a_{j}(\frac{C_{j,m}(\psi_{i}\lambda_{\tau^{j,m}}^{\prime})}{\sqrt{2k}}-\frac{\int_{S^{*}_{F_{0}}\Sigma}\psi_{i}\lambda_{\tau^{j,m}}^{\prime}\wedge
d\lambda_{\tau^{j,m}}^{\prime}}{\sqrt{\mathrm{vol}_{0}(\lambda_{\tau^{j,m}}^{\prime})}})|\leq\check{c}(\hat{c}+D).$

As $m\rightarrow+\infty$, $\lambda_{\tau^{j,m}}^{\prime}$ converges to
$\lambda_{\tau^{\infty}}^{\prime}$ and the Reeb orbits in
$\\{C_{j,m}\\}_{m\geq 1}$ have bounded periods. Therefore, up to a
subsequence, we can obtain a limiting current

$C_{\infty}:=(2k)^{-\frac{1}{2}}\mathrm{vol}_{0}(\lambda_{\tau^{\infty}}^{\prime})^{\frac{1}{2}}\lim_{m\rightarrow+\infty}\sum_{j=1}^{N+1}a_{j}C_{j,m}\in
L(S^{*}_{F_{0}}\Sigma,\lambda_{\tau^{\infty}}^{\prime}),$

and then from (4.5), we have

$\displaystyle|C_{\infty}(\psi_{i}\lambda_{\tau^{\infty}}^{\prime})-\int_{S^{*}_{F_{0}}\Sigma}\psi_{i}\lambda_{\tau^{\infty}}^{\prime}\wedge
d\lambda_{\tau^{\infty}}^{\prime}|\leq\mathrm{vol}_{0}(\lambda_{\tau^{\infty}}^{\prime})^{\frac{1}{2}}\cdot\check{c}(\hat{c}+D).$

For any given $\lambda_{0}\in\mathcal{U}$, after fixing $\hat{c}$ such that
$\lambda_{[0,1]^{N}}\subset\mathcal{U}$, one can choose $\check{c}$
sufficiently small by perturbing the corresponding family $F_{[0,1]^{N}}$ of
$\lambda_{[0,1]^{N}}$ to $F_{[0,1]^{N}}^{\prime}$, properly. Finally, applying
Lemma 3.2, we can further perturb $F_{\tau^{\infty}}^{\prime}$ to
$F_{\tau^{\infty}}^{\prime\prime}$ (resp. $\lambda_{\tau^{\infty}}^{\prime}$
to $\lambda_{\tau^{\infty}}^{\prime\prime}$) sufficiently small, such that
$C_{\infty}\in L(S^{*}_{F_{0}}\Sigma,\lambda_{\tau^{\infty}}^{\prime\prime})$
is nondegenerate, then $\lambda_{\tau^{\infty}}^{\prime\prime}$ and
$C_{\infty}$ are what we need.

At last, the proof is the same to the case
$\mathcal{U}\subset\Lambda^{sFin}(S^{*}_{F}\Sigma,\xi_{F})$, since the
$[0,1]^{N}$-family (4.2) is a fiberwised scaling of
$\lambda_{0}\in\Lambda^{sFin}(S^{*}_{F}\Sigma,\xi_{F})$ on $S^{*}_{F}\Sigma$,
which is still a family of $\Lambda^{sFin}(S^{*}_{F}\Sigma,\xi_{F})$. The rest
of steps above are also available in this case. Therefore, Claim 4 holds. ∎

Finally, we complete the proof of Theorem 1.1 for
$\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$. The case of
$\Lambda^{sFin}(S^{*}_{F}\Sigma,\xi_{F})$ is completely the same. Denote by
$\Lambda^{Fin}(N,\epsilon)$ the collection of contact forms
$\lambda\in\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$, which possesses a
nondegenerate Reeb 1-current $C_{\lambda,\epsilon}\in
L(S^{*}_{F}\Sigma,\lambda)$ such that
$|C_{\lambda,\epsilon}(\psi_{i}\lambda)-\int_{S^{*}_{F}\Sigma}\psi_{i}\lambda\wedge
d\lambda|<\epsilon$ for any $i=1,\cdots,N$.

To show $\Lambda^{Fin}(N,\epsilon)$ is open and dense. The denseness follows
from Claim 4. To show the openness. Let
$C_{\lambda_{0},\epsilon}=\sum_{j=1}^{k}a_{j}\gamma_{j}$ be the corresponding
nondegenerate Reeb 1-current of $\lambda_{0}\in\Lambda^{Fin}(N,\epsilon)$. For
any $\lambda$ nearby $\lambda_{0}$ in
$\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$, there is a nondegenerate Reeb
1-current $C_{\lambda,\epsilon}=\sum_{j=1}^{k}a_{j}\gamma_{j}(\lambda)\in
L(S^{*}_{F}\Sigma,\lambda)$, which is continuously dependent of $\lambda$,
since a nondegenerate Reeb orbit varies smoothly under a sufficiently small
perturbation of Reeb flow. Therefore, the openness follows.

Now, we choose $\epsilon=N^{-1}$. Then the intersection

$\Lambda^{Fin}_{0}(S^{*}_{F}\Sigma,\xi_{F}):=\cap_{N\geq
1}\Lambda^{Fin}(N,N^{-1})$

is of second category in $\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$. By
definition, for any $\lambda\in\Lambda^{Fin}_{0}(S^{*}_{F}\Sigma,\xi_{F})$,
there is a nondegenerate sequence $\\{C_{\lambda,N^{-1}}\\}_{N\geq 1}\subset
L(S^{*}_{F}\Sigma,\lambda)$ such that

$|C_{\lambda,N^{-1}}(\psi_{i}\lambda)-\int_{S^{*}_{F}\Sigma}\psi_{i}\lambda\wedge
d\lambda|\rightarrow 0,\ \forall i=1,\cdots,N\rightarrow+\infty.$

Since $\\{\psi_{i}\\}_{i\geq 1}$ is $C^{0}$-dense in
$C^{\infty}(\Sigma,\mathbb{R})$, therefore

$|C_{\lambda,N^{-1}}(f\lambda)-\int_{S^{*}_{F}\Sigma}f\lambda\wedge
d\lambda|\rightarrow 0$

for any $f\in C^{\infty}(\Sigma,\mathbb{R})$ by continuity. Therefore, Theorem
1.1 follows. ∎

###### The proof of Theorem 1.2..

Take a sequence of $\\{\psi_{i}\\}_{i\geq 1}$ in
$C^{\infty}(\Sigma,\mathbb{R})\subset C^{\infty}(S^{*}_{g}\Sigma,\mathbb{R})$,
which is $C^{0}$-dense in $C^{\infty}(\Sigma,\mathbb{R})$. For any open subset
$\mathcal{U}\subset\Lambda^{Rie}(S^{*}_{g}\Sigma,\xi_{g})$, and any element
$\lambda_{0}\in\mathcal{U}$, we can define a $[0,1]^{N}$-family of contact
forms (similar to (4.2)) as

(4.6) $\lambda_{\tau}=e^{f_{\tau}}\lambda(g_{0}),\
f_{\tau}=\frac{1}{\hat{c}}\sum_{i=1}^{N}\tau_{i}\psi_{i}\in
C^{\infty}(\Sigma,\mathbb{R}),\ \forall\tau\in[0,1]^{N}.$

By Proposition 2.5 (2), we have
$(S^{*}_{g_{0}}\Sigma,\lambda_{\tau})\cong(S^{*}_{g_{\tau}}\Sigma,\lambda(g_{\tau}))$,
where $g_{[0,1]^{N}}\in\mathcal{M}^{N}(\Sigma,g)$ is defined by

$g_{\tau}(x):=\exp(\frac{2}{\hat{c}}\sum_{i=1}^{N}\tau_{i}\psi_{i}(x))g_{0}(x),\
\forall x\in\Sigma.$

By choosing $\hat{c}$ sufficiently large, we have
$\lambda_{[0,1]^{N}}\subset\mathcal{U}$. Then by following the proof of Claim
4, for any $N\in\mathbb{N}_{+}$ and $\epsilon>0$, we can also find a contact
form $\lambda\in\mathcal{U}$ and a nondegenerate Reeb 1-current
$C_{\lambda,\epsilon}\in L(S^{*}_{g_{0}}\Sigma,\lambda)$ such that

(4.7)
$|C_{\lambda,\epsilon}(\psi_{i}\lambda)-\int_{S^{*}_{g}\Sigma}\psi_{i}\lambda\wedge
d\lambda|<\epsilon,\ \forall i=1,\cdots,N.$

Then we define $\Lambda^{Rie}(N,\epsilon)$ as the collection of contact forms
$\lambda\in\Lambda^{Rie}(S^{*}_{g}\Sigma,\xi_{g})$, which possesses a
nondegenerate Reeb 1-current $C_{\lambda,\epsilon}\in
L(S^{*}_{g}\Sigma,\lambda)$ satisfying (4.7). Similar to
$\Lambda^{Fin}(N,\epsilon)$, one can also show that
$\Lambda^{Rie}(N,\epsilon)$ is open and dense. Therefore, the intersection

$\Lambda^{Rie}_{0}(S^{*}_{g}\Sigma,\xi_{g}):=\cap_{N\geq
1}\Lambda^{Rie}(N,N^{-1}),$

is of second category in $\Lambda^{Rie}(S^{*}_{g}\Sigma,\xi_{g})$. By
definition, for any $\lambda\in\Lambda^{Rie}_{0}(S^{*}_{g}\Sigma,\xi_{g})$,
there is a nondegenerate sequence $\\{C_{\lambda,N^{-1}}\\}_{N\geq 1}\subset
L(S^{*}_{g}\Sigma,\lambda)$ satisfies

$|C_{\lambda,N^{-1}}(\psi_{i}\lambda)-\int_{S^{*}_{g}\Sigma}\psi_{i}\lambda\wedge
d\lambda|\rightarrow 0,\ \forall i=1,\cdots,N\rightarrow+\infty.$

Since $\\{\psi_{i}\\}_{i\geq 1}$ is $C^{0}$-dense in
$C^{\infty}(\Sigma,\mathbb{R})$, then by continuity, we have

$|C_{\lambda,N^{-1}}(f\lambda)-\int_{S^{*}_{g}\Sigma}f\lambda\wedge
d\lambda|\rightarrow 0,\ \forall f\in C^{\infty}(\Sigma,\mathbb{R}).$

Therefore, Theorem 1.2 follows. ∎

###### The proof of Theorem 1.3..

We only prove the Finsler case, and the symmetric Finsler case is similar.
Assume $\lambda\in\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$ satisfies (1.6),
then we can identify $(S^{*}_{F}\Sigma,\lambda)$ with
$(S^{*}_{F_{\lambda}}\Sigma,\lambda(F_{\lambda}))$ for some
$F_{\lambda}\in\mathcal{F}(\Sigma)$. By Proposition 2.4 (4), each element
$\sigma\in\mathrm{CG}(\Sigma,F_{\lambda})$ is one to one corresponding to
$\gamma_{\sigma}\in R(S^{*}_{F}\Sigma,\lambda)$.

Firstly, we rewrite $C_{i}=\sum_{j=1}^{m_{i}}a_{i,j}\gamma_{\sigma_{i,j}}\in
L(S^{*}_{F}\Sigma,\lambda)$, which can be approximated by a Reeb 1-current
$C_{i}^{\prime}=\frac{1}{n_{i}}\sum_{j=1}^{m_{i}}r_{\sigma_{i,j}}\gamma_{\sigma_{i,j}}$
with $\mathbb{Q}$-coefficients for any $i\geq 1$. After choosing $f\equiv 1$
and then replacing $C_{i}$ by $C_{i}^{\prime}$ for any $i\geq 1$, (1.6)
becomes

$\lim_{i\rightarrow+\infty}\frac{1}{n_{i}}\sum_{j=1}^{m_{i}}r_{\sigma_{i,j}}l_{F_{\lambda}}(\sigma_{i,j})=2\pi\mathrm{vol}(\Sigma,F_{\lambda}),$

where $l_{F_{\lambda}}(\sigma_{i,j})=\gamma_{\sigma_{i,j}}(\lambda)$ is the
length of $\sigma_{i,j}$ and
$2\pi\mathrm{vol}(\Sigma,F_{\lambda})=\mathrm{vol}(S^{*}_{F}\Sigma,\lambda)$
by Proposition 2.4 (6) and (8). Moreover, since

$\gamma_{\sigma_{i,j}}(f\lambda)=\sigma_{i,j}(f\beta(F_{\lambda})),\
\int_{S^{*}_{F}\Sigma}f\lambda\wedge
d\lambda=2\pi\int_{\Sigma}fd\mathrm{vol}_{F_{\lambda}},\ \forall f\in
C^{\infty}(\Sigma,\mathbb{R}),$

by Proposition 2.4 (5) and (7), (1.6) can be further written as

(4.8)
$\displaystyle\lim_{i\rightarrow+\infty}\frac{\sum_{j=1}^{m_{i}}r_{\sigma_{i,j}}\sigma_{i,j}(f\beta(F_{\lambda}))}{\sum_{j=1}^{m_{i}}r_{\sigma_{i,j}}l_{F_{\lambda}}(\sigma_{i,j})}=\frac{\int_{\Sigma}fd\mathrm{vol}_{F_{\lambda}}}{\mathrm{vol}(\Sigma,F_{\lambda})},\
\forall f\in C^{\infty}(\Sigma,\mathbb{R}).$

Finally, we denote by $\mathcal{A}_{N}=\sum_{i=1}^{N}n_{i}A_{i}$,
$\mathcal{B}_{N}=\sum_{i=1}^{N}n_{i}B_{i}$ and
$C(f)=\frac{d\mathrm{vol}_{F_{\lambda}}(f)}{\mathrm{vol}(\Sigma,F_{\lambda})}$,
where $n_{i}\in\mathbb{N}_{+}$ and
$A_{i}=\sum_{j=1}^{m_{i}}r_{\sigma_{i,j}}\sigma_{i,j}$,
$B_{i}=\sum_{j=1}^{m_{i}}r_{\sigma_{i,j}}l_{F_{\lambda}}(\sigma_{i,j})$.

######  Claim 5.

$\lim_{N\rightarrow+\infty}\frac{\mathcal{A}_{N}(f)}{\mathcal{B}_{N}}=C(f)$,
where $n_{N}$ satisfies
$n_{N}\frac{B_{N}}{\mathcal{B}_{N-1}}\rightarrow+\infty$.

Indeed, write $\mathcal{A}_{N_{0}+1,N}=\mathcal{A}_{N}-\mathcal{A}_{N_{0}}$
and $\mathcal{B}_{N_{0}+1,N}=\mathcal{B}_{N}-\mathcal{B}_{N_{0}}$, we have

$\displaystyle\frac{\mathcal{A}_{N}(f)}{\mathcal{B}_{N}}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}_{N_{0}}(f)+\mathcal{A}_{N_{0}+1,N}(f)}{\mathcal{B}_{N_{0}}+\mathcal{B}_{N_{0}+1,N}}$
$\displaystyle=$
$\displaystyle\frac{\mathcal{A}_{N_{0}}(f)}{\mathcal{B}_{N_{0}}}\frac{\mathcal{B}_{N_{0}}}{\mathcal{B}_{N_{0}}+\mathcal{B}_{N_{0}+1,N}}+\frac{\mathcal{A}_{N_{0}+1,N}(f)}{\mathcal{B}_{N_{0}+1,N}}\frac{\mathcal{B}_{N_{0}+1,N}}{\mathcal{B}_{N_{0}}+\mathcal{B}_{N_{0}+1,N}}$
$\displaystyle\leq$
$\displaystyle\frac{\mathcal{A}_{N_{0}}(f)}{\mathcal{B}_{N_{0}}}\frac{\mathcal{B}_{N_{0}}}{\mathcal{B}_{N_{0}}+\mathcal{B}_{N_{0}+1,N}}+(\epsilon_{N_{0}}(f)+C(f))\frac{\mathcal{B}_{N_{0}+1,N}}{\mathcal{B}_{N_{0}}+\mathcal{B}_{N_{0}+1,N}}$
$\displaystyle\leq$
$\displaystyle\frac{\mathcal{A}_{N_{0}}(f)}{\mathcal{B}_{N_{0}}}+\epsilon_{N_{0}}(f)+C(f)=:D_{N_{0}}(f),$

where $|\frac{A_{N}(f)}{B_{N}}-C(f)|<\epsilon_{N_{0}}(f),\ \forall N>N_{0}$ by
(4.8). Then $\frac{\mathcal{A}_{N}(f)}{\mathcal{B}_{N}}$ is bounded and

$\displaystyle\frac{\mathcal{A}_{N}(f)}{\mathcal{B}_{N}}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{A}_{N-1}(f)+n_{N}A_{N}(f)}{\mathcal{B}_{N-1}+n_{N}B_{N}}$
$\displaystyle=$
$\displaystyle\frac{\mathcal{A}_{N-1}(f)}{\mathcal{B}_{N-1}}\frac{1}{1+n_{N}\frac{B_{N}}{\mathcal{B}_{N-1}}}+\frac{A_{N}(f)}{B_{N}}\frac{n_{N}\frac{B_{N}}{\mathcal{B}_{N-1}}}{1+n_{N}\frac{B_{N}}{\mathcal{B}_{N-1}}}$
$\displaystyle\rightarrow$
$\displaystyle\lim_{N\rightarrow+\infty}\frac{A_{N}(f)}{B_{N}}=C(f).$

Therefore, the Claim 5 implies (1.8), where the sequence
$\\{\sigma_{i}\\}_{i\geq 1}$ is chosen as $\cup_{i\geq
1}\\{\sigma_{i,j}^{n_{i}r_{\sigma_{i,j}}}\\}_{j=1}^{m_{i}}$ in
$\mathrm{CG}(\Sigma,F_{\lambda})$ and $\sigma^{r}$ denotes the $r$-iteration
of $\sigma$. Finally, since $\mathcal{F}(\Sigma)$ is homeomorphic to
$\Lambda^{Fin}(S^{*}_{F}\Sigma,\xi_{F})$ by definition (1.3), then
$F_{\lambda}$ is the generic choice of $\mathcal{F}(\Sigma)$. ∎

###### The proof of Theorem 1.4..

The proof is similar to Theorem 1.3. Assume $\lambda$ is a generic element in
$\Lambda^{Rie}(S^{*}_{g}\Sigma,\xi_{g})$ with satisfing (1.7), then we can
identify $(S^{*}_{g}\Sigma,\lambda)$ with
$(S^{*}_{g_{\lambda}}\Sigma,\lambda(g_{\lambda}))$ for some
$g_{\lambda}\in\mathcal{M}(\Sigma)$ and each closed geodesic
$\sigma\in\mathrm{CG}(\Sigma,g_{\lambda})$ is one to one corresponding to
$\gamma_{\sigma}\in R(S^{*}_{g}\Sigma,\lambda)$ by Proposition 2.5 (4).

Firstly, we rewrite $C_{i}=\sum_{j=1}^{m_{i}}a_{i,j}\gamma_{\sigma_{i,j}}\in
L(S^{*}_{g}\Sigma,\lambda)$. After approximating $C_{i}$ by
$\mathbb{Q}$-coefficient Reeb 1-current
$C_{i}^{\prime}=\frac{1}{n_{i}}\sum_{j=1}^{m_{i}}r_{\sigma_{i,j}}\gamma_{\sigma_{i,j}}$
for any $i\geq 1$ and using (1.7) with $f\equiv 1$, we have

$\lim_{i\rightarrow+\infty}\frac{1}{n_{i}}\sum_{j=1}^{m_{i}}r_{\sigma_{i,j}}l_{g_{\lambda}}(\sigma_{i,j})=2\pi\mathrm{vol}(\Sigma,g_{\lambda}),$

where $l_{g_{\lambda}}(\sigma_{i,j})=\gamma_{\sigma_{i,j}}(\lambda)$ and
$2\pi\mathrm{vol}(\Sigma,g_{\lambda})=\mathrm{vol}(S^{*}_{g}\Sigma,\lambda)$
by Proposition 2.5 (6) and (8). Moreover, let $s(g)$ denote the arc length
parameter along any $\sigma\in\mathrm{CG}(\Sigma,g)$. By using Proposition 2.5
(5) and (7), we have

$\gamma_{\sigma_{i,j}}(f\lambda)=\sigma_{i,j}(fds(g_{\lambda})),\
\int_{S^{*}_{g}\Sigma}f\lambda\wedge
d\lambda=2\pi\int_{\Sigma}fd\mathrm{vol}_{g_{\lambda}},\ \forall f\in
C^{\infty}(\Sigma,\mathbb{R}).$

Then (1.7) can be further written as

(4.9)
$\displaystyle\lim_{i\rightarrow+\infty}\frac{\sum_{j=1}^{m_{i}}r_{\sigma_{i,j}}\sigma_{i,j}(fds(g_{\lambda}))}{\sum_{j=1}^{m_{i}}r_{\sigma_{i,j}}l_{g_{\lambda}}(\sigma_{i,j})}=\frac{\int_{\Sigma}fd\mathrm{vol}_{g_{\lambda}}}{\mathrm{vol}(\Sigma,g_{\lambda})},\
\forall f\in C^{\infty}(\Sigma,\mathbb{R}).$

Finally, by Claim 5 and (4.9), we obtain (1.9), where the sequence
$\\{\sigma_{i}\\}_{i\geq 1}$ is chosen as $\cup_{i\geq
1}\\{\sigma_{i,j}^{n_{i}r_{\sigma_{i,j}}}\\}_{j=1}^{m_{i}}$ in
$\mathrm{CG}(\Sigma,g_{\lambda})$. Since $\mathcal{M}(\Sigma)$ is homeomorphic
to $\Lambda^{Rie}(S^{*}_{g}\Sigma,\xi_{g})$ by definition (1.4), then
$g_{\lambda}$ is the generic choice of $\mathcal{M}(\Sigma)$. ∎

##  5\. Appendix

###  5.1. Contact Local model

In this subsection, we establish the following special local model by using
the classical Moser trick in the Contact Darboux theorem, which is needed in
Section 3. To the idea of this proof, we refer to Theorem 2.5.2 and Example
2.5.16 in [13].

######  Proposition 5.1.

Let $(Y^{2n+1},\alpha)$ be a contact manifold and $\gamma$ is a Reeb orbit
with minimal period $T$. There exists a tubular neighborhood $U$ of
$\gamma(\mathbb{R})$ and a diffeomorphism $\Phi:S^{1}\times
D^{2n}_{\epsilon}\rightarrow U((t,0)\mapsto\gamma(Tt))$ such that
$\Phi^{*}\alpha=H\alpha_{st}$ for some smooth function $H:S^{1}\times
D^{2n}_{\epsilon}\rightarrow\mathbb{R}_{+}$ with $H(t,0)=T$ and $dH(t,0)=0$
and $H\circ\Phi|_{U_{1}}\equiv T$, where
$\alpha_{st}=dz+\sum_{i=1}^{n}x_{i}dy_{i}$ and $U_{1}\subset U$ is a smaller
disk-like neighborhood of $\gamma(t_{0})$ for some $t_{0}\in[0,T]$.

###### Proof.

Firstly, we can choose a diffeomorphism $\Phi_{0}:S^{1}\times
D^{2n}_{\epsilon}\rightarrow U$ with sending $(t,0)$ to $\gamma(Tt)$. Let
$\alpha_{1}=\Phi^{*}_{0}\alpha$, then $\alpha_{1}(\frac{\partial}{\partial
z})=T$ and the Reeb vector field $R_{1}$ of $\alpha_{1}$ becomes
$T^{-1}\frac{\partial}{\partial z}$ on $S^{1}\times\\{0\\}$. Since
$\ker\alpha_{1}|_{S^{1}\times\\{0\\}}$ transverses to $S^{1}\times\\{0\\}$,
therefore, we can deform $\Phi_{0}$ such that
$\ker\alpha_{1}|_{(t,0)}=\\{t\\}\times\mathbb{R}^{2n}$. Moreover, since the
symplectic bundle over $S^{1}\times\\{0\\}$ is unique up to isomorphism, then
there is an bundle isomorphism

$\Psi:(S^{1}\times\mathbb{R}^{2n},Td\alpha_{st})\rightarrow(S^{1}\times\mathbb{R}^{2n},d\alpha_{1}),\
\Psi^{*}d\alpha_{1}=Td\alpha_{st}.$

By replacing $\Phi_{0}$ to $\Phi_{0}\circ\Psi$, we have
$\alpha_{1}=T\alpha_{st}$ and
$d\alpha_{1}|_{\mathbb{R}^{2n}}=Td\alpha_{st}|_{\mathbb{R}^{2n}}$ on
$S^{1}\times\\{0\\}$. Now we choose a function $H\in C^{\infty}(S^{1}\times
D^{2n}_{\epsilon})$ such that $H|_{S^{1}\times\\{0\\}}=T$,
$dH|_{S^{1}\times\\{0\\}}=0$ and $H|_{U_{1}}\equiv T$ which always exists.
Then on $S^{1}\times\\{0\\}$, we have

$\iota_{R_{1}}d(H\alpha_{st})=H\iota_{R_{1}}d\alpha_{st}=\iota_{\frac{\partial}{\partial
z}}d\alpha_{st}=0.$

Therefore, we have $H\alpha_{st}=\alpha_{1}$ and $d(H\alpha_{st})=d\alpha_{1}$
on $S^{1}\times\\{0\\}$.

Let $\alpha_{0}=H\alpha_{st}$. Next, we extend such a relation to $U$ by Moser
trick. Let $\alpha_{t}=(1-t)\alpha_{0}+t\alpha_{1}$ and we aim to find a
vector field $X_{t}$ on $S^{1}\times D^{2n}_{\epsilon}$ such that the induced
flow $\phi_{t}$ satisfies

(5.1) $\phi_{t}^{*}\alpha_{t}=\alpha_{0},\ \forall t\in[0,1],\ \mathrm{and\
fixing}\ S^{1}\times\\{0\\}.$

Differentiate (5.1) by $t$, we obtain that
$\phi_{t}^{*}(\alpha_{t}+\mathcal{L}_{X_{t}}\alpha_{t})=0$, i.e. $X_{t}$
solves

$\dot{\alpha}_{t}+d(\alpha_{t}(X_{t}))+l_{X_{t}}d\alpha_{t}=0.$

Since $\alpha_{t}\wedge d\alpha_{t}=\alpha_{1}\wedge d\alpha_{1}$ on
$S^{1}\times\\{0\\}$, then if $\epsilon$ is sufficient small, $\alpha_{t}$ are
still contact forms on $S^{1}\times D^{2n}_{\epsilon}$. Let $R_{t}$ be the
Reeb vector field of $\alpha_{t}$ and write $X_{t}=H_{t}R_{t}+Y_{t}$ with
$Y_{t}\in\ker\alpha_{t}$ and $H_{t}\in C^{\infty}(\\{t\\}\times
D^{2n}_{\epsilon})$. Since we need $X_{t}|_{S^{1}\times\\{0\\}}=0$, we also
require $H_{t}(0)=0$. Then $Y_{t}$ satisfies

$\dot{\alpha}_{t}+dH_{t}+l_{Y_{t}}d\alpha_{t}=0.$

Finally, by choosing $dH_{t}|_{S^{1}\times\\{0\\}}=0$, the nondegenercy of
$d\alpha_{t}$ implies that $Y_{t}$ exists once $H_{t}$ is given, and $Y_{t}=0$
on $S^{1}\times\\{0\\}$. Therefore, we obtain the desired $X_{t}$ in
$S^{1}\times D^{2n}_{\epsilon}$ and $\phi_{t}$ exists. Then
$H\alpha_{st}=(\Phi_{0}\circ\phi_{1})^{*}\alpha$ on $S^{1}\times
S^{2n}_{\epsilon}$ for $\epsilon$ sufficiently small. Then
$\Phi=\Phi_{0}\circ\phi_{1}$ is what we want. ∎

###  5.2. Sard-Smale theorem

In this subsection, we briefly introduce the Sard-Smale theorem, which will be
needed in Section 3. To the details, we refer to [26], Section 1.3.5 of [6]
and Appendix A in [24]. A _Fredholm operator_ is a continuous linear map
$L:E_{1}\rightarrow E_{2}$ between Banach spaces with satisfying $\dim\ker
L<+\infty,\ \dim\mathrm{coker}L<\infty$ and closed image. The _Fredholm index_
of $L$ is defined by

$\mathrm{Ind}L=\dim\ker L-\dim\mathrm{coker}L\in\mathbb{Z}.$

A _Fredholm map_ is a $C^{1}$ map $f:X\rightarrow Y$ between connected Banach
manifolds such that $Df(x):T_{x}X\rightarrow T_{f(x)}Y$ is Fredholm for any
$x\in X$. The Fredholm index of $f$ is defined as
$\mathrm{Ind}f:=\mathrm{Ind}Df(x)$ for some $x$, which is independent of $x$.
A point $x\in X$ is called _singular point_ of $f$ if $Df(x)$ is not
surjective and $f(x)\in Y$ is called a _singular value_ of $f$. $x\in X$ is
called _regular point_ if $x$ is not singular and $y\in Y$ is called a
_regular value_ of $f$ if $y$ is not a singular value.

We firstly introduce the famous Sard theorem in [25] (c.f. [26]).

######  Lemma 5.1.

[Theorem 1.2 in [26]] Let $M,N$ be $m,n$-dimensional manifold and $U\subset M$
is an open submanifold. If $f:U\rightarrow N$ is a $C^{r}$-map, where $r\geq
1$ and $r>\max\\{0,m-n\\}$, then the set of singular values of $f$ has measure
zero in $N$.

The Sard theorem was generalized to infinite dimensional case by Smale in [26]
(c.f. [6]) as follows.

######  Lemma 5.2.

[Theorem 1.3.16 in [6]] Suppose $X$ is a separable Banach space, $U\subset X$
is an open set and $Y$ is a Banach space. Let $f:U\rightarrow Y$ be a Fredholm
$C^{r}$-map, where $r>\max\\{0,\mathrm{Ind}f\\}$, then the regular set of $f$
is of second category.

There is also a infinite dimensional version of implicit function theorem as
follows:

######  Lemma 5.3.

[Theorem A.3.3 in [24]] Let $X,Y$ be Banach space and $U\subset X$ is an open
subset. If $f:U\rightarrow Y$ is a $C^{r}$-map and $y$ is a regular value of
$f$, then $f^{-1}(y)\subset X$ is a $C^{r}$-submanifold. Furthermore, if $f$
is Fredholm, then $\dim f^{-1}(y)=\mathrm{Ind}f$.

Let $f:X\rightarrow Y$ be a $C^{1}$ map and let $W\subset Y$ be a submanifold.
$f$ is said to be _transversal_ to submanifold $W$ if

$\mathrm{Im}Df(x)+T_{f(x)}W=T_{f(x)}Y,\ x\in f^{-1}(W).$

Then as a corollary of Lemma 5.2 and 5.3, we have

######  Lemma 5.4.

[Theorem 3.3 in [26]] Let $f:X\rightarrow Y$ be a $C^{r}$-map which is
transversal to a finitely dimensional smooth submanifold $W\subset Y$, then
$f^{-1}(W)$ is a $C^{r}$-submanifold of $X$. Furthermore, if $f$ is Fredholm,
then

$\dim f^{-1}(W)=\mathrm{Ind}f+\dim W.$

###  5.3. Lipschitz function with small variation

In this subsection, we introduce a result for Lipschitz functions, which is
proved by Marques, Neves and Song in [23] and needed in Section 4. Let
$f:[0,1]^{N}\rightarrow\mathbb{R}$ be any fixed Lipschitz function with $\max
f-\min f\leq 2\epsilon$ and let $\mathcal{A}\subset[0,1]^{N}$ be any fixed
full measure subset. We have the following result.

######  Lemma 5.5.

(Lemma 3 in [23]) For any $\delta>0$ and $N\in\mathbb{N}_{+}$, there exist a
$\epsilon=\epsilon(\delta,N)>0$ and $N+1$ many sequences
$\\{\tau^{j,m}\\}_{m\geq 1}$ in $\mathcal{A}$, such that

  * (1)

$\\{\tau^{j,m}\\}_{m\geq 1}$ converges to a point $\tau^{\infty}\in(0,1)^{N}$
for any $j=1,\cdots,N+1$.

  * (2)

$f$ is differentiable at $\tau^{j,m}$ for any $m\geq 1$ and $j=1,\cdots,N+1$.

  * (3)

$\\{\nabla f(\tau^{j,m})\\}_{m\geq 1}$ converges to some $v_{j}$ for any $j$,
which satisfies

$d_{\mathbb{R}^{N}}(0,\mathrm{conv}(v_{1},\cdots,v_{N+1}))<\delta.$

where $d_{\mathbb{R}^{N}}$ is the Euclidean distance in $\mathbb{R}^{N}$.

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