# On the minimal number of closed geodesics on positively curved Finsler
spheres

Huagui Duan  (Huagui Duan) School of Mathematical Sciences and LPMC, Nankai
University, Tianjin 300071, The People’s Republic of China
<EMAIL_ADDRESS>and  Dong Xie  (Dong Xie) School of Mathematical
Sciences, Nankai University, Tianjin 300071, The People’s Republic of China
<EMAIL_ADDRESS>
(Date: 2024-05-19)

###### Abstract.

In this paper, we proved that for every Finsler metric on $S^{n}$ $(n\geq 4)$
with reversibility $\lambda$ and flag curvature $K$ satisfying
$(\frac{2n-3}{n-1})^{2}(\frac{\lambda}{\lambda+1})^{2}<K\leq 1$ and
$\lambda<\frac{n-1}{n-2}$, there exist at least $n$ prime closed geodesics on
$(S^{n},F)$, which solved a conjecture of Katok and Anosov for such positivley
curved spheres when $n$ is even. Furthermore, if the number of closed
geodesics on such positively curved Finsler $S^{n}$ is finite, then there
exist at least $2\left[\frac{n}{2}\right]-1$ non-hyperbolic closed geodesics.

###### Key words and phrases:

Closed geodesics, positively curved, Finsler sphere, index theory, Morse
theory.

###### 2010 Mathematics Subject Classification:

53C22, 53C60, 58E10

H. Duan is partially supported by National Key R&D Program of China (Grant No.
2020YFA0713300) and NNSFC (Nos. 12271268 and 12361141812) and the Fundamental
Research Funds for the Central Universities. D. Xie is partially supported by
NNSFC (No. 12361141812).

##  1\. Introduction and main results

###  1.1. Introduction

This paper is devoted to searching for the minimal number of distinct closed
geodesics on positively curved Finsler spheres. This problem is closely
related to a conjecture of Katok and Anosov, which claims that every Finsler
metric on a sphere $S^{2k}$ or $S^{2k-1}$ carries at least $2k$ distinct
closed geodesics. In fact, in 1973, Katok constructed certain irreversible
Finsler metrics on spheres with flag curvature $K=1$, which have exactly $2k$
distinct closed geodesics. And based on these examples, in 1974 Anosov
proposed the above conjecture.

A closed curve in a Finsler manifold is a closed geodesic if it is locally the
shortest path connecting any two nearby points on this curve. A closed
geodesic $c:S^{1}=\mathbb{R}/\mathbb{Z}\to M$ is prime if it is not a multiple
covering (i.e., iteration) of any other closed geodesics. Here the $m$-th
iteration $c^{m}$ of $c$ is defined by $c^{m}(t)=c(mt)$. The inverse curve
$c^{-1}$ of $c$ is defined by $c^{-1}(t)=c(1-t)$ for $t\in\mathbb{R}$. We call
two prime closed geodesics $c$ and $d$ distinct if there is no
${\theta}\in(0,1)$ such that $c(t)=d(t+{\theta})$ for all $t\in\mathbb{R}$. We
shall omit the word distinct when we talk about more than one prime closed
geodesic. For a reversible Finsler (or Riemannian) metric, two closed
geodesics $c$ and $d$ are called geometrically distinct if $c(S^{1})\neq
d(S^{1})$.

For a closed geodesic $c$ on $(M,\,F)$ with $\dim M=n$, denote by $P_{c}$ the
linearized Poincaré map of $c$. Then $P_{c}\in{\rm Sp}(2n-2)$ is a symplectic
matrix. We define the elliptic height  $e(P_{c})$ of $P_{c}$ to be the total
algebraic multiplicity of all eigenvalues of $P_{c}$ on the unit circle
$\mathbb{U}=\\{z\in\mathbb{C}|\;|z|=1\\}$ in the complex plane $\mathbb{C}$.
Since $P_{c}$ is symplectic, $e(P_{c})$ is even and $0\leq e(P_{c})\leq 2n-2$.
Then $c$ is called hyperbolic if $e(P_{c})=0$; elliptic if $e(P_{c})=2n-2$;
non-degenerate if $1$ is not an eigenvalue of $P_{c}$. A Finsler metric $F$ on
$M$ is called bumpy if all the closed geodesics (including iterations) on
$(M,F)$ are non-degenerate.

There is a famous conjecture in geometry which claims the existence of
infinitely many closed geodesics on a compact Riemannian manifold. This
conjecture has been proved for many cases, but sitll open for CROSS (compact
rank one symmetric spaces) except for $S^{2}$ (cf. [Ban], [DLW1], [Fra], [GM],
[Hin2], [Hin3] and therein). It was surprising when Katok in [Kat] found
certain irreversible Finsler metrics on CROSS (including spheres) with only
finitely many closed geodesics and all of them are non-degenerate and elliptic
(cf. [Zil]).

Based on the Katok’s examples, in 1974, Anosov in [Ano] proposed the following
conjecture (also cf. [Lon3] and [BuM]), and we call the Katok-Anosov
conjecture:

$\displaystyle{\mathcal{N}}(S^{n},F)\geq
2\left[\frac{n+1}{2}\right]\quad\mbox{for any Finsler metric $F$ on}\ S^{n},$
(1.1)

where, denote by ${\mathcal{N}}(M,F)$ the number of distinct closed geodesics
on $(M,F)$, and $[a]=\max\\{k\in\mathbb{Z}\,|\,k\leq a\\}$. In 2010, Bangert
and Long in [BaL] proved this conjecture for every Finsler $2$-dimensional
sphere $(S^{2},F)$. Since then, the index iteration theory of closed geodesics
(cf. [Bot] and [Lon2]) has been applied to study the closed geodesic problem
on Finsler manifolds.

When $n\geq 3$, the Katok-Anosov conjecture is still open in full of
generality. About the multiplicity and stability problem of closed geodesics,
two classes of typical conditions, including the positively curved condition
and the non-degenerate (or bumpy) condition, have been studied widely.

In [Rad4], Rademacher has introduced the reversibility $\lambda=\lambda(M,F)$
of a compact Finsler manifold $(M,F)$, which is defined by

$\displaystyle\lambda=\max\\{F(-X)\ |\ X\in TM,\ F(X)=1\\}\geq 1.$

Then Rademacher in [Rad5] has obtained some results about the multiplicity and
stability of closed geodesics. For example, let $F$ be a Finsler metric on
$S^{n}$ with reversibility ${\lambda}$ and flag curvature $K$ satisfying
$\left(\frac{{\lambda}}{1+{\lambda}}\right)^{2}<K\leq 1$, then there exist at
least $n/2-1$ closed geodesics with length $<2n\pi$. If
$\left(\frac{2n-4}{n-1}\right)^{2}\left(\frac{{\lambda}}{1+{\lambda}}\right)^{2}<K\leq
1$ with ${\lambda}<\frac{n-1}{n-3}$ and $n\geq 4$, then there exist at least
$n-2$ closed geodesics. If $\frac{9{\lambda}^{2}}{4(1+{\lambda})^{2}}<K\leq 1$
with ${\lambda}<2$, then there exists a closed geodesic of elliptic-parabolic,
i.e., its linearized Poincaré map split into $2$-dimensional rotations and a
part whose eigenvalues are $\pm 1$.

In 2012, Wang in [Wan2] proved the conjecture (1.1) for $(S^{n},F)$ provided
that $F$ is bumpy and its flag curvature $K$ satisfies
$\left(\frac{\lambda}{1+\lambda}\right)^{2}<K\leq 1$. Also in [Wan2], Wang
showed that for every bumpy Finsler metric $F$ on $S^{n}$ satisfying
$\frac{9{\lambda}^{2}}{4(1+{\lambda})^{2}}<K\leq 1$, there exist two prime
elliptic closed geodesics if ${\mathcal{N}}(S^{n},F)<+\infty$. As a further
generalization, Duan, Long and Wang in [DLW2] obtained the optimal lower bound
of the number of distinct closed geodesics on every CROSS $(M,F)$ (including
spheres) if $F$ is bumpy and the flag curvature is non-negative.

Without the assumption of the non-degenerate (or bumpy) metric, the proof of
conjecture (1.1) is extremely hard. For example, under the assumption of the
positively curved condition
$\left(\frac{{\lambda}}{1+{\lambda}}\right)^{2}<K\leq 1$, it can only be
showed that ${\mathcal{N}}(S^{n},F)\geq 3$ (cf. [Dua1]) or ([Dua2])) and
${\mathcal{N}}(S^{n},F)\geq\left[\frac{n+1}{2}\right]$ (cf. [Wan4]), where
$n\geq 3$.

###  1.2. Main results

Motivated by the conjecture (1.1) and the above results, in this paper, we
further consider the positively curved Finsler $n$-dimensional sphere
$(S^{n},F)$ without the bumpy assumption, and obtain some multiplicity results
of closed geodesics on $(S^{n},F)$. In particular, the following Theorem 1.1
solved the Katok-Anosov conjecture (1.1) for certain positively curved spheres
$(S^{n},F)$ when $n$ is even, and this result is optimal due to the Katok’s
examples.

Next we assume that the reversibility $\lambda$ and flag curvature $K$ of
$(S^{n},F)$ satisfies the following condition:

$\displaystyle\left(\frac{2n-3}{n-1}\right)^{2}\left(\frac{\lambda}{\lambda+1}\right)^{2}<K\leq
1,\quad\mbox{where}\ \lambda<\frac{n-1}{n-2}.$ (1.2)

Theorem 1.1. Every Finsler metric $F$ on $S^{n}$ $(n\geq 4)$ satisfying (1.2)
carries at least $n$ prime closed geodesics.

On the other hand, we also obtain some stability results of closed geodesics
on spheres satisfying (1.2) as follows.

Theorem 1.2. Every Finsler metric $F$ on $S^{n}$ $(n\geq 4)$ satisfying (1.2)
carries at least $2\left[\frac{n}{2}\right]-1$ non-hyperbolic prime closed
geodesics if ${\mathcal{N}}(S^{n},F)<+\infty$.

Theorem 1.3. Every Finsler metric $F$ on $S^{n}$ $(n\geq 4)$ satisfying (1.2)
carries at least $n-2$ prime closed geodesics, whose linearized Poincaré map
possesses at least an eigenvalue of the form $\exp(\pi i\mu)$ with an
irrational $\mu$, if ${\mathcal{N}}(S^{n},F)<+\infty$.

When the curvature condition is weaker than (1.2), we can obtain the existence
of at least $n-1$ prime closed geodesics, i.e. we obtained the following
result.

Theorem 1.4. On a Finsler sphere $(S^{n},F)$ $(n\geq 4)$ satisfying
$\frac{9}{4}\left(\frac{\lambda}{\lambda+1}\right)^{2}<K\leq 1$ and
$\lambda<2$, there exist at least $n-1$ prime closed geodesics.

###  1.3. Outline of the proof

The main ingredients of the proof of Theorem 1.1 include the Fadell-Rabinowitz
index theory (see Lemma 2.5 and Lemma 2.6 below) from [Rad3] and [Wan3], the
Morse theory and the enhanced common index jump theorem (see Theorem 3.6
below) recently developed in [DLW2], which originally grown from the
groundbreaking work [LoZ].

Let $(S^{n},F)$ be a Finsler sphere satisfying (1.2) and assume that there are
only finitely many prime closed geodesics on $(S^{n},F)$, which is denoted by
$\\{c_{j}\\}_{j=1}^{p}$. The outline of the proof of Theorem 1.1 is as
follows.

First, under the curvature condition (1.2), Theorem 1.1 in [Rad4] established
the lower bound of the length of any prime closed geodesics on $(S^{n},F)$,
which in turn gives the lower bound of $i(c_{j}^{m})$ and the average index
$\hat{i}(c_{j})$ for each closed geodesic $c_{j}$ (cf. Lemma 4.1 below).

Second, it follows from the enhanced common index jump theorem (cf. Theorem
3.6 below) that there exist an appropriately chosen $(p+1)$-tuple
$(N,m_{1},\cdots,m_{p})\in\mathbb{N}^{p+1}$ such that the indices
$i(c_{j}^{2m_{j}\pm m})$ of iterates around $2m_{j}$ of all $c_{j}$’s satisfy
some precise equalities and estimates (cf. (4.6)-(4.9)). Then, together with
those index estimates from Lemma 4.1, we can get much more useful and crucial
estimates (cf. (4.15)-(4.20) and Lemma 4.2) for later proofs.

Third, with these above preparations about index estimates, as the first step
of the proof of Theorem 1.1, we established the existence of $n-1$ prime
closed geodesics. More precisely, for each $i\in
G_{1}\equiv[N-(n-2),N-1]\cap\mathbb{N}$ (and denoted by
$\bar{G_{1}}\equiv[2N-(n-3),2N+(n-3)]\cap\mathbb{N}$), by the Fadell-
Rabinowitz index theory (cf. Lemma 2.6), there exists some
$j(i)\in\\{1,\cdots,p\\}$ and $m(i)\in\mathbb{N}$ such that the integer
$2i+n-1$ (belonging to $\bar{G_{1}}$) lies between $i(c_{j(i)}^{m(i)})$ and
$i(c_{j(i)}^{m(i)})+\nu(c_{j(i)}^{m(i)})$, and $c_{j(i)}^{m(i)}$ has non-
trivial critical module $\overline{C}_{2i+n-1}(E,c_{j(i)}^{m(i)})\neq 0$.
Through the analysis of index estimates and Morse theory, it can be deduced
that $m(i)=2m_{j(i)}$ (cf. Claim 1 below). Then the assumption
${\mathcal{N}}(S^{n},F)<+\infty$ and Lemma 2.5 yield that the different
integer $i\in G_{1}$ corresponds to the distinct closed geodesic $c_{j(i)}$.
Thus it shows that $p\geq\\#[N-(n-2),N-1]=n-2$. In addition, by a result of
[Dua1] or [Dua2] (cf. Lemma 4.3 below), we can find one more closed geodesic
$c_{j_{0}}$ satisfying $\overline{C}_{q}(E,c_{j_{0}}^{2m_{j_{0}}})\neq 0$ only
for $q=2N+n-1$. Notice that $2N+n-1\notin\bar{G_{1}}$, thus $c_{j_{0}}$ is
distinct from each of the $n-2$ closed geodesics found above. This yields
$p={\mathcal{N}}(S^{n},F)\geq n-1$.

Finally, as the second step of the proof of Theorem 1.1, we carry out the
proof by contradiction, i.e., assuming that there are only $n-1$ closed
geodesics. In order to get a contradiction, here we consider another integer
interval $G_{2}\equiv[N-(2n-3),N-(n-1)]\cap\mathbb{N}$ with empty intersection
with $G_{1}$ (and denoted by
$\bar{G_{2}}\equiv[2N-3(n-1)+2,2N-(n-1)]\cap\mathbb{N}$). Then, by index
estimates and Morse theory, we can show that there holds $m(i)=2m_{j(i)}-1$
for every integer $i\in G_{2}$ and $\\{c_{j(i)}\,|\,i\in G_{2}\\}$ is the one-
to-one correspondence to the $n-1$ distinct closed geodesics found in Step 1
(cf. Claim 2 and (4.46) below). Then it can be deduced that the Morse-type
number $M_{2N-(n-1)}$ only can be contributed by
$c_{j(N-n+1)}^{2m_{j(N-n+1)}-1}$ (cf. Claim 3 and (4.54) below). However, this
is a contradiction $1=M_{2N-(n-1)}\geq b_{2N-(n-1)}=2$ by the Morse
inequality, where note that $N$ can be chosen to be the multiple of $n-1$.
Thus we obtain the existence of another distinct closed geodesic, i.e.
${\mathcal{N}}(S^{n},F)\geq n$.

In this paper, let $\mathbb{N}$, $\mathbb{N}_{0}$, $\mathbb{Z}$, $\mathbb{Q}$,
$\mathbb{R}$, and $\mathbb{C}$ denote the sets of natural integers, non-
negative integers, integers, rational numbers, real numbers, and complex
numbers respectively. We use only singular homology modules with
$\mathbb{Q}$-coefficients. For an $S^{1}$-space $X$, we denote by
$\overline{X}$ the quotient space $X/S^{1}$. We define the functions

$\left\\{\begin{array}[]{ll}[a]=\max\\{k\in\mathbb{Z}\,|\,k\leq
a\\},&E(a)=\min\\{k\in\mathbb{Z}\,|\,k\geq
a\\},\cr\varphi(a)=E(a)-[a],&\\{a\\}=a-[a].\cr\end{array}\right.$ (1.3)

Especially, $\varphi(a)=0$ if $a\in\mathbb{Z}\,$, and $\varphi(a)=1$ if
$a\notin\mathbb{Z}\,$.

##  2\. Critical point theory and Fadell-Rabinowitz index theory

###  2.1. Critical point theory for closed geodesics

On a compact Finsler manifold $(M,F)$, we choose an auxiliary Riemannian
metric. This endows the space $\Lambda=\Lambda M$ of $H^{1}$-maps
$\gamma:S^{1}\rightarrow M$ with a natural Riemannian Hilbert manifold
structure on which the group $S^{1}=\mathbb{R}/\mathbb{Z}$ acts continuously
by isometries, cf. [Kli]. This action is defined by translating the parameter,
i.e., $(s\cdot\gamma)(t)=\gamma(t+s)$ for all $\gamma\in{\Lambda}$ and $s,t\in
S^{1}$. The Finsler metric $F$ defines an energy functional $E$ and a length
functional $L$ on $\Lambda$ by

$E(\gamma)=\frac{1}{2}\int_{S^{1}}F(\dot{\gamma}(t))^{2}dt,\quad
L(\gamma)=\int_{S^{1}}F(\dot{\gamma}(t))dt.$ (2.1)

Both functionals are invariant under the $S^{1}$-action. The functional $E$ is
$C^{1,1}$ on ${\Lambda}$ and satisfies the Palais-Smale condition (cf. [Mer]).
Thus we can apply the deformation theorems in [Cha] and [MaW]. The critical
points of $E$ of positive energies are precisely the closed geodesics
$c:S^{1}\to M$ of the Finsler structure. If $c\in\Lambda$ is a closed geodesic
then $c$ is a regular curve, i.e., $\dot{c}(t)\not=0$ for all $t\in S^{1}$,
and this implies that the second differential $E^{\prime\prime}(c)$ of $E$ at
$c$ exists. As usual we define the index $i(c)$ of $c$ as the maximal
dimension of subspaces of $T_{c}\Lambda$ on which $E^{\prime\prime}(c)$ is
negative definite, and the nullity $\nu(c)$ of $c$ so that $\nu(c)+1$ is the
dimension of the null space of $E^{\prime\prime}(c)$.

For $m\in\mathbb{N}$ we denote the $m$-fold iteration map
$\phi^{m}:\Lambda\rightarrow\Lambda$ by

$\phi^{m}({\gamma})(t)={\gamma}(mt)\qquad\forall\,{\gamma}\in{\Lambda},t\in
S^{1}.$ (2.2)

We also use the notation $\phi^{m}(\gamma)=\gamma^{m}$. For a closed geodesic
$c$, the average index is defined by

$\hat{i}(c)=\lim_{m\rightarrow\infty}\frac{i(c^{m})}{m}.$ (2.3)

If $\gamma\in\Lambda$ is not constant then the multiplicity $m(\gamma)$ of
$\gamma$ is the order of the isotropy group $\\{s\in S^{1}\mid
s\cdot\gamma=\gamma\\}$. If $m(\gamma)=1$ then $\gamma$ is called prime. Hence
$m(\gamma)=m$ if and only if there exists a prime curve
$\tilde{\gamma}\in\Lambda$ such that $\gamma=\tilde{\gamma}^{m}$.

In this paper for ${\kappa}\in\mathbb{R}$ and a closed geodesic $c$ we denote
by

${\Lambda}^{{\kappa}}=\\{d\in{\Lambda}\,|\,E(d)\leq{\kappa}\\},\qquad{\Lambda}(c)=\\{{\gamma}\in{\Lambda}\mid
E({\gamma})<E(c)\\}.$ (2.4)

We call a closed geodesic satisfying the isolation condition, if the following
holds:

(Iso) For all $m\in\mathbb{N}$ the orbit $S^{1}\cdot c^{m}$ is an isolated
critical orbit of $E$.

Note that if the number of prime closed geodesics on a Finsler manifold is
finite, then all the closed geodesics satisfy (Iso).

Using singular homology with rational coefficients we consider the following
critical $\mathbb{Q}$-module of a closed geodesic $c\in\Lambda$:

$\overline{C}_{*}(E,c)=H_{*}\left(({\Lambda}(c)\cup S^{1}\cdot
c)/S^{1},{\Lambda}(c)/S^{1}\right).$ (2.5)

Proposition 2.1. (cf. Satz 6.11 of [Rad2] or Proposition 3.12 of [BaL]) Let
$c$ be a prime closed geodesic on a Finsler manifold $(M,F)$ satisfying (Iso).
Then we have

$\displaystyle\overline{C}_{q}(E,c^{m})$ $\displaystyle\equiv
H_{q}\left(({\Lambda}(c^{m})\cup S^{1}\cdot
c^{m})/S^{1},{\Lambda}(c^{m})/S^{1}\right)$
$\displaystyle=\left(H_{i(c^{m})}(U_{c^{m}}^{-}\cup\\{c^{m}\\},U_{c^{m}}^{-})\otimes
H_{q-i(c^{m})}(N_{c^{m}}^{-}\cup\\{c^{m}\\},N_{c^{m}}^{-})\right)^{+\mathbb{Z}_{m}}$

(i) When $\nu(c^{m})=0$, there holds

$\displaystyle\overline{C}_{q}(E,c^{m})=\left\\{\begin{array}[]{ll}\mathbb{Q},&\quad{\it
if}\;\;i(c^{m})-i(c)\in 2\mathbb{Z},\;{\it and}\;q=i(c^{m}),\cr 0,&\quad{\it
otherwise}.\cr\end{array}\right.$ (2.8)

(ii) When $\nu(c^{m})>0$, there holds

$\displaystyle\overline{C}_{q}(E,c^{m})=H_{q-i(c^{m})}(N_{c^{m}}^{-}\cup\\{c^{m}\\},N_{c^{m}}^{-})^{(-1)^{i(c^{m})-i(c)}\mathbb{Z}_{m}},$

where $N_{c^{m}}$ is a local characteristic manifold at $c^{m}$ and
$N^{-}_{c^{m}}=N_{c^{m}}\cap\Lambda(c^{m})$, $U_{c^{m}}$ is a local negative
disk at $c^{m}$ and $U^{-}_{c^{m}}=U_{c^{m}}\cap\Lambda(c^{m})$,
$H_{\ast}(X,A)^{\pm\mathbb{Z}_{m}}=\\{[\xi]\in
H_{\ast}(X,A)\,|\,T_{\ast}[\xi]=\pm[\xi]\\}$ where $T$ is a generator of the
$\mathbb{Z}_{m}$-action.

Denote by

$\displaystyle
k_{j}(c^{m})\equiv\dim\,H_{j}(N_{c^{m}}^{-}\cup\\{c^{m}\\},N_{c^{m}}^{-})^{(-1)^{i(c^{m})-i(c)}\mathbb{Z}_{m}}.$
(2.9)

Clearly the integers $k_{j}(c^{m})$ equal to $0$ when $j<0$ or $j>\nu(c^{m})$
and can take only values $0$ or $1$ when $j=0$ or $j=\nu(c^{m})$.

Proposition 2.2. (cf. Satz 6.13 of [Rad2]) Let $c$ be a prime closed geodesic
on a Finsler manifold $(M,F)$ satisfying (Iso). For any $m\in\mathbb{N}$, we
have

(i) If $k_{0}(c^{m})=1$, there holds $k_{j}(c^{m})=0$ for $1\leq
j\leq\nu(c^{m})$.

(ii) If $k_{\nu(c^{m})}(c^{m})=1$, there holds $k_{j}(c^{m})=0$ for $0\leq
j\leq\nu(c^{m})-1$.

(iii) If $k_{j}(c^{m})\geq 1$ for some $1\leq j\leq\nu(c^{m})-1$, there holds
$k_{\nu(c^{m})}(c^{m})=0=k_{0}(c^{m})$.

Set $\overline{{\Lambda}}^{0}=\overline{\Lambda}^{0}M=\\{{\rm
constant\;point\;curves\;in\;}M\\}\cong M$. Let $(X,Y)$ be a space pair such
that the Betti numbers $b_{i}=b_{i}(X,Y)=\dim H_{i}(X,Y;\mathbb{Q})$ are
finite for all $i\in\mathbb{Z}$. As usual the Poincaré series of $(X,Y)$ is
defined by the formal power series $P(X,Y)=\sum_{i=0}^{\infty}b_{i}t^{i}$. We
need the following results on Betti numbers.

Theorem 2.3. (cf. [Hin1], Theorem 2.4 and Remark 2.5 of [Rad1]) We have the
Poincaré series of $(\overline{{\Lambda}}S^{n},\overline{{\Lambda}}^{0}S^{n})$
as the following:

(i) When $n=2k+1$ is odd

$\displaystyle
P(\overline{{\Lambda}}S^{n},\overline{{\Lambda}}^{0}S^{n})(t)=t^{n-1}\left(\frac{1}{1-t^{2}}+\frac{t^{n-1}}{1-t^{n-1}}\right)=t^{2k}\left(\frac{1}{1-t^{2}}+\frac{t^{2k}}{1-t^{2k}}\right).$
(2.10)

Thus for $q\in\mathbb{Z}$ and $l\in\mathbb{N}_{0}$, we have

$\displaystyle{b}_{q}$ $\displaystyle=$
$\displaystyle{b}_{q}(\overline{{\Lambda}}S^{n},\overline{{\Lambda}}^{0}S^{n})={\rm
rank}H_{q}(\overline{{\Lambda}}S^{n},\overline{{\Lambda}}^{0}S^{n})$ (2.14)
$\displaystyle=$ $\displaystyle\;\;\left\\{\begin{array}[]{ll}2,&\quad{\it
if}\quad q\in\\{4k+2l,\quad l=0\;{\rm mod}\;k\\},\cr 1,&\quad{\it if}\quad
q\in\\{2k\\}\cup\\{2k+2l,\quad l\neq 0\;{\rm mod}\;k\\},\cr 0&\quad{\it
otherwise}.\cr\end{array}\right.$

(ii) When $n=2k$ is even

$\displaystyle P(\overline{{\Lambda}}S^{n},\overline{{\Lambda}}^{0}S^{n})(t)$
$\displaystyle=$ $\displaystyle
t^{n-1}\left(\frac{1}{1-t^{2}}+\frac{t^{n(m+1)-2}}{1-t^{n(m+1)-2}}\right)\frac{1-t^{nm}}{1-t^{n}}$
(2.15) $\displaystyle=$ $\displaystyle
t^{2k-1}\left(\frac{1}{1-t^{2}}+\frac{t^{4k-2}}{1-t^{4k-2}}\right),$

where $m=1$ by Theorem 2.4 of [Rad1]. Thus for $q\in\mathbb{Z}$ and
$l\in\mathbb{N}_{0}$, we have

$\displaystyle{b}_{q}$ $\displaystyle=$
$\displaystyle{b}_{q}(\overline{{\Lambda}}S^{n},\overline{{\Lambda}}^{0}S^{n})={\rm
rank}H_{q}(\overline{{\Lambda}}S^{n},\overline{{\Lambda}}^{0}S^{n})$ (2.19)
$\displaystyle=$ $\displaystyle\;\;\left\\{\begin{array}[]{ll}2,&\quad{\it
if}\quad q\in\\{6k-3+2l,\quad l=0\;{\rm mod}\;2k-1\\},\cr 1,&\quad{\it
if}\quad q\in\\{2k-1\\}\cup\\{2k-1+2l,\quad l\neq 0\;{\rm mod}\;2k-1\\},\cr
0&\quad{\it otherwise}.\cr\end{array}\right.$

We have the following version of the Morse inequality.

Theorem 2.4. (cf. Theorem 6.1 of [Rad2]) Suppose that there exist only
finitely many prime closed geodesics $\\{c_{j}\\}_{1\leq j\leq p}$ on $(M,F)$,
and $0\leq a<b\leq\infty$. Define for each $q\in\mathbb{Z}$,

$\displaystyle{M}_{q}(\overline{{\Lambda}}^{b},\overline{{\Lambda}}^{a})$
$\displaystyle=$ $\displaystyle\sum_{1\leq j\leq p,\;a<E(c^{m}_{j})<b}{\rm
rank}{\overline{C}}_{q}(E,c^{m}_{j})$
$\displaystyle{b}_{q}(\overline{{\Lambda}}^{b},\overline{{\Lambda}}^{a})$
$\displaystyle=$ $\displaystyle{\rm
rank}H_{q}(\overline{{\Lambda}}^{b},\overline{{\Lambda}}^{a}).$

Then there holds

$\displaystyle M_{q}(\overline{{\Lambda}}^{b},\overline{{\Lambda}}^{a})$
$\displaystyle-$ $\displaystyle
M_{q-1}(\overline{{\Lambda}}^{b},\overline{{\Lambda}}^{a})+\cdots+(-1)^{q}M_{0}(\overline{{\Lambda}}^{b},\overline{{\Lambda}}^{a})$
(2.20) $\displaystyle\geq$ $\displaystyle
b_{q}(\overline{{\Lambda}}^{b},\overline{{\Lambda}}^{a})-b_{q-1}(\overline{{\Lambda}}^{b},\overline{{\Lambda}}^{a})+\cdots+(-1)^{q}b_{0}(\overline{{\Lambda}}^{b},\overline{{\Lambda}}^{a}),$
$\displaystyle{M}_{q}(\overline{{\Lambda}}^{b},\overline{{\Lambda}}^{a})$
$\displaystyle\geq$
$\displaystyle{b}_{q}(\overline{{\Lambda}}^{b},\overline{{\Lambda}}^{a}).$
(2.21)

###  2.2. Fadell-Rabinowitz index theory

Next we recall the Fadell-Rabinowitz index in a relative version due to
[Rad3]. Let $X$ be an $S^{1}$-space, $A\subset X$ a closed $S^{1}$-invariant
subset. Note that the cup product defines a homomorphism

$H^{\ast}_{S^{1}}(X)\otimes H^{\ast}_{S^{1}}(X,\;A)\rightarrow
H^{\ast}_{S^{1}}(X,\;A):\quad(\zeta,\;z)\rightarrow\zeta\cup z,$ (2.22)

where $H_{S^{1}}^{\ast}$ is the $S^{1}$-equivariant cohomology with rational
coefficients in the sense of Borel (cf. Chapter IV of [Bor]). Fix a
characteristic class $\eta\in H^{2}(CP^{\infty})$. Let
$f^{\ast}:H^{\ast}(CP^{\infty})\rightarrow H_{S^{1}}^{\ast}(X)$ be the
homomorphism induced by a classifying map $f:X_{S^{1}}\rightarrow
CP^{\infty}$. Now for $\gamma\in H^{\ast}(CP^{\infty})$ and $z\in
H^{\ast}_{S^{1}}(X,\;A)$, let $\gamma\cdot z=f^{\ast}(\gamma)\cup z$. Then the
order $\mbox{ord}_{\eta}(z)$ with respect to $\eta$ is defined by

$\mbox{ord}_{\eta}(z)=\inf\\{k\in\mathbb{N}\cup\\{\infty\\}\;|\;\eta^{k}\cdot
z=0\\}.$ (2.23)

By Proposition 3.1 of [Rad3], there is an element $z\in
H^{n+1}_{S^{1}}(\Lambda,\;\Lambda^{0})$ of infinite order, i.e.,
$\mbox{ord}_{\eta}(z)=\infty$. For $\kappa\geq 0$, we denote by
$j_{\kappa}:(\Lambda^{\kappa},\;\Lambda^{0})\rightarrow(\Lambda\;\Lambda^{0})$
the natural inclusion and define the function $d_{z}:\mathbb{R}^{\geq
0}\rightarrow\mathbb{N}\cup\\{\infty\\}$:

$d_{z}(\kappa)=\mbox{ord}_{\eta}(j_{\kappa}^{\ast}(z)).$ (2.24)

Denote by $d_{z}(\kappa-)=\lim_{\epsilon\searrow 0}d_{z}(\kappa-\epsilon)$,
where $t\searrow a$ means $t>a$ and $t\to a$.

By the Section 5 of [Rad3], the function $d_{z}$ is non-decreasing and
$\lim_{\lambda\searrow\kappa}d_{z}(\lambda)=d_{z}(\kappa).$ Each discontinuous
point of $d_{z}$ is a critical value of the energy functional $E$. If
$d_{z}(\kappa)-d_{z}(\kappa-)\geq 2$, then there are infinitely many prime
closed geodesics $c$ with energy $\kappa$. So, for each $i\geq 1$, we define

$\kappa_{i}=\inf\\{\delta\in\mathbb{R}\;|\>d_{z}(\delta)\geq i\\}.$

Then we have the following:

Lemma 2.5. (cf. Lemma 2.3 of [Wan3]) Suppose there are only finitely many
prime closed geodesics on $(S^{n},\,F)$. Then each $\kappa_{i}$ is a critical
value of $E$. If $\kappa_{i}=\kappa_{j}$ for some $i<j$, then there are
infinitely many prime closed geodesics on $(S^{n},\,F)$.

Lemma 2.6. (cf. Lemma 2.4 of [Wan3]) Suppose there are only finitely many
prime closed geodesics on $(S^{n},\,F)$. Then for every $i\in\mathbb{N}$,
there exists a closed geodesic $c$ on $(S^{n},\,F)$ such that

$\displaystyle E(c)=\kappa_{i},\quad\overline{C}_{2i+\dim(z)-2}(E,c)\neq 0.$
(2.25)

Definition 2.7. A prime closed geodesic $c$ is $(m,i)$-variationally visible,
if there exist some $m,i\in\mathbb{N}$ such that (2.25) holds for $c^{m}$ and
$\kappa_{i}$. We call $c$ infinitely variationally visible, if there exist
infinitely many $m,i\in\mathbb{N}$ such that $c$ is $(m,i)$-variationally
visible. Denote by $\mathcal{V}_{\infty}(S^{n},F)$ the set of infinitely
variationally visible closed geodesics.

Theorem 2.8. (cf. Theorem 2.6 of [Wan3]) Suppose there are only finitely many
prime closed geodesics on $(S^{n},\,F)$. Then for any
$c\in\mathcal{V}_{\infty}(S^{n},F)$, there holds

$\frac{\hat{i}(c)}{L(c)}=2\sigma.$ (2.26)

where
$\sigma=\lim\inf_{i\rightarrow\infty}i/\sqrt{2\kappa_{i}}=\lim\sup_{i\rightarrow\infty}{i}/{\sqrt{2\kappa_{i}}}$.

##  3\. Index iteration theory for closed geodesics

In this section, we recall briefly the index theory for symplectic paths
developed by Long and his coworkers, and all the details can be found in
[Lon2].

Let $c$ be a closed geodesic on an orientable Finsler manifold $M=(M,\,F)$.
Denote the linearized Poincaré map of $c$ by $P_{c}\in{\rm Sp}(2n-2)$. Then
$P_{c}$ is a symplectic matrix. Note that the index iteration formulae in
[Lon1] of 2000 (cf. Chap. 8 of [Lon2]) work for Morse indices of iterated
closed geodesics (cf. [LLo], Chap. 12 of [Lon2]). Since every closed geodesic
on $M$ is orientable. Then by Theorem 1.1 of [Liu] of Liu (cf. also [Wil]),
the initial Morse index of a closed geodesic $c$ on $M$ coincides with the
index of a corresponding symplectic path introduced by Conley, Zehnder and
Long in 1984-1990 (cf. [Lon2]).

As usual, the symplectic group ${\rm Sp}(2n)$ is defined by

${\rm Sp}(2n)=\\{M\in{\rm GL}(2n,\mathbb{R})\,|\,M^{T}JM=J\\},$

whose topology is induced from that of $\mathbb{R}^{4n^{2}}$. For $\tau>0$ we
are interested in ${\rm Sp}(2n)$:

${\mathcal{P}}_{\tau}(2n)=\\{{\gamma}\in C([0,\tau],{\rm
Sp}(2n))\,|\,{\gamma}(0)=I_{2n}\\},$

which is equipped with the topology induced from that of ${\rm Sp}(2n)$. The
following real function was introduced in [Lon1]:

$D_{{\omega}}(M)=(-1)^{n-1}\overline{{\omega}}^{n}\det(M-{\omega}I_{2n}),\qquad\forall{\omega}\in\mathbb{U},\,M\in{\rm
Sp}(2n).$

Thus for any ${\omega}\in\mathbb{U}$ the following hypersurface in ${\rm
Sp}(2n)$ is defined in [Lon1]:

${\rm Sp}(2n)_{{\omega}}^{0}=\\{M\in{\rm Sp}(2n)\,|\,D_{{\omega}}(M)=0\\}.$

For any $M\in{\rm Sp}(2n)_{{\omega}}^{0}$, we define a co-orientation of ${\rm
Sp}(2n)_{{\omega}}^{0}$ at $M$ by the positive direction
$\frac{d}{dt}Me^{t{\epsilon}J}|_{t=0}$ of the path $Me^{t{\epsilon}J}$ with
$0\leq t\leq 1$ and small ${\epsilon}>0$. Let

$\displaystyle{\rm Sp}(2n)_{{\omega}}^{\ast}$ $\displaystyle=$
$\displaystyle{\rm Sp}(2n)\setminus{\rm Sp}(2n)_{{\omega}}^{0},$
$\displaystyle{\mathcal{P}}_{\tau,{\omega}}^{\ast}(2n)$ $\displaystyle=$
$\displaystyle\\{{\gamma}\in{\mathcal{P}}_{\tau}(2n)\,|\,{\gamma}(\tau)\in{\rm
Sp}(2n)_{{\omega}}^{\ast}\\},$
$\displaystyle{\mathcal{P}}_{\tau,{\omega}}^{0}(2n)$ $\displaystyle=$
$\displaystyle{\mathcal{P}}_{\tau}(2n)\setminus{\mathcal{P}}_{\tau,{\omega}}^{\ast}(2n).$

For any two continuous arcs $\xi$ and $\eta:[0,\tau]\to{\rm Sp}(2n)$ with
$\xi(\tau)=\eta(0)$, we define

$\eta\ast\xi(t)=\left\\{\begin{array}[]{ll}\xi(2t),&\quad{\rm if}\;0\leq
t\leq\tau/2,\cr\eta(2t-\tau),&\quad{\rm if}\;\tau/2\leq
t\leq\tau.\cr\end{array}\right.$

Given any two $2m_{k}\times 2m_{k}$ matrices of square block form
$M_{k}=\left(\begin{array}[]{cc}A_{k}&B_{k}\cr
C_{k}&D_{k}\cr\end{array}\right)$ with $k=1,2$, as in [Lon2], the
$\;\diamond$-product of $M_{1}$ and $M_{2}$ is defined by the following
$2(m_{1}+m_{2})\times 2(m_{1}+m_{2})$ matrix $M_{1}\diamond M_{2}$:

$M_{1}\diamond M_{2}=\left(\begin{array}[]{cccc}A_{1}&0&B_{1}&0\cr
0&A_{2}&0&B_{2}\cr C_{1}&0&D_{1}&0\cr 0&C_{2}&0&D_{2}\cr\end{array}\right).$

Denote by $M^{\diamond k}$ the $k$-fold $\diamond$-product
$M\diamond\cdots\diamond M$. Note that the $\diamond$-product of any two
symplectic matrices is symplectic. For any two paths
${\gamma}_{j}\in{\mathcal{P}}_{\tau}(2n_{j})$ with $j=0$ and $1$, let
${\gamma}_{0}\diamond{\gamma}_{1}(t)={\gamma}_{0}(t)\diamond{\gamma}_{1}(t)$
for all $t\in[0,\tau]$.

A special path $\xi_{n}$ is defined by

$\xi_{n}(t)=\left(\begin{array}[]{cc}2-\frac{t}{\tau}&0\cr
0&(2-\frac{t}{\tau})^{-1}\cr\end{array}\right)^{\diamond n}\qquad{\rm
for}\;0\leq t\leq\tau.$ (3.1)

Definition 3.1. (cf. [Lon1], [Lon2]) For any ${\omega}\in\mathbb{U}$ and
$M\in{\rm Sp}(2n)$, define

$\nu_{{\omega}}(M)=\dim_{\mathbb{C}}\ker_{\mathbb{C}}(M-{\omega}I_{2n}).$
(3.2)

For any $\tau>0$ and ${\gamma}\in{\mathcal{P}}_{\tau}(2n)$, define

$\nu_{{\omega}}({\gamma})=\nu_{{\omega}}({\gamma}(\tau)).$ (3.3)

If ${\gamma}\in{\mathcal{P}}_{\tau,{\omega}}^{\ast}(2n)$, define

$i_{{\omega}}({\gamma})=[{\rm Sp}(2n)_{{\omega}}^{0}:{\gamma}\ast\xi_{n}],$
(3.4)

where the right hand side of (3.4) is the usual homotopy intersection number,
and the orientation of ${\gamma}\ast\xi_{n}$ is its positive time direction
under homotopy with fixed end points.

If ${\gamma}\in{\mathcal{P}}_{\tau,{\omega}}^{0}(2n)$, we let
$\mathcal{F}({\gamma})$ be the set of all open neighborhoods of ${\gamma}$ in
${\mathcal{P}}_{\tau}(2n)$, and define

$i_{{\omega}}({\gamma})=\sup_{U\in\mathcal{F}({\gamma})}\inf\\{i_{{\omega}}(\beta)\,|\,\beta\in
U\cap{\mathcal{P}}_{\tau,{\omega}}^{\ast}(2n)\\}.$ (3.5)

Then

$(i_{{\omega}}({\gamma}),\nu_{{\omega}}({\gamma}))\in\mathbb{Z}\times\\{0,1,\ldots,2n\\},$

is called the index function of ${\gamma}$ at ${\omega}$.

For any symplectic path ${\gamma}\in{\mathcal{P}}_{\tau}(2n)$ and
$m\in\mathbb{N}$, we define its $m$-th iteration
${\gamma}^{m}:[0,m\tau]\to{\rm Sp}(2n)$ by

${\gamma}^{m}(t)={\gamma}(t-j\tau){\gamma}(\tau)^{j},\qquad{\rm for}\quad
j\tau\leq t\leq(j+1)\tau,\;j=0,1,\ldots,m-1.$ (3.6)

We still denote the extended path on $[0,+\infty)$ by ${\gamma}$.

Definition 3.2. (cf. [Lon1], [Lon2]) For any
${\gamma}\in{\mathcal{P}}_{\tau}(2n)$, we define

$(i({\gamma},m),\nu({\gamma},m))=(i_{1}({\gamma}^{m}),\nu_{1}({\gamma}^{m})),\qquad\forall
m\in\mathbb{N}.$ (3.7)

The average index $\hat{i}({\gamma},m)$ per $m\tau$ for $m\in\mathbb{N}$ is
defined by

$\hat{i}({\gamma},m)=\lim_{k\to+\infty}\frac{i({\gamma},mk)}{k}.$ (3.8)

For $M\in{\rm Sp}(2n)$ and ${\omega}\in\mathbb{U}$, the splitting numbers
$S_{M}^{\pm}({\omega})$ of $M$ at ${\omega}$ are defined by

$S_{M}^{\pm}({\omega})=\lim_{{\epsilon}\to
0^{+}}i_{{\omega}\exp(\pm\sqrt{-1}{\epsilon})}({\gamma})-i_{{\omega}}({\gamma}),$
(3.9)

for any path ${\gamma}\in{\mathcal{P}}_{\tau}(2n)$ satisfying
${\gamma}(\tau)=M$.

For a given path $\gamma\in{\mathcal{P}}_{\tau}(2n)$, we deform it to a new
path $\eta$ in ${\mathcal{P}}_{\tau}(2n)$ so that

$i_{1}(\gamma^{m})=i_{1}(\eta^{m}),\quad\nu_{1}(\gamma^{m})=\nu_{1}(\eta^{m}),\quad\forall
m\in{\mathbb{N}},$ (3.10)

and that $(i_{1}(\eta^{m}),\nu_{1}(\eta^{m}))$ is easy enough to compute. This
leads to finding homotopies $\delta:[0,1]\times[0,\tau]\to{\rm Sp}(2n)$
starting from $\gamma$ in ${\mathcal{P}}_{\tau}(2n)$ and keeping the end
points of the homotopy always stay in a certain suitably chosen maximal subset
of ${\rm Sp}(2n)$ so that (3.10) always holds. In fact, this set was first
discovered in [Lon1] as the path connected component $\Omega^{0}(M)$
containing $M=\gamma(\tau)$ of the set

$\displaystyle\Omega(M)=\\{N\in{\rm Sp}(2n)\,$ $\displaystyle|$
$\displaystyle\,\sigma(N)\cap{\mathbb{U}}=\sigma(M)\cap{\mathbb{U}}\;{\rm
and}\;$ (3.11)
$\displaystyle\qquad\nu_{\lambda}(N)=\nu_{\lambda}(M),\;\forall\,\lambda\in\sigma(M)\cap{\mathbb{U}}\\}.$

Here $\Omega^{0}(M)$ is called the homotopy component of $M$ in ${\rm
Sp}(2n)$.

In [Lon1] and [Lon2], the following symplectic matrices were introduced as
basic normal forms:

$\displaystyle D(\lambda)=\left(\begin{array}[]{cc}{\lambda}&0\cr
0&{\lambda}^{-1}\cr\end{array}\right),$ $\displaystyle{\lambda}=\pm 2,$ (3.14)
$\displaystyle N_{1}({\lambda},b)=\left(\begin{array}[]{cc}{\lambda}&b\cr
0&{\lambda}\cr\end{array}\right),$ $\displaystyle{\lambda}=\pm 1,b=\pm 1,0,$
(3.17) $\displaystyle
R({\theta})=\left(\begin{array}[]{cc}\cos{\theta}&-\sin{\theta}\cr\sin{\theta}&\cos{\theta}\cr\end{array}\right),$
$\displaystyle{\theta}\in(0,\pi)\cup(\pi,2\pi),$ (3.20) $\displaystyle
N_{2}({\omega},b)=\left(\begin{array}[]{cc}R({\theta})&b\cr
0&R({\theta})\cr\end{array}\right),$
$\displaystyle{\theta}\in(0,\pi)\cup(\pi,2\pi),$ (3.23)

where $b=\left(\begin{array}[]{cc}b_{1}&b_{2}\cr
b_{3}&b_{4}\cr\end{array}\right)$ with $b_{i}\in\mathbb{R}$ and
$b_{2}\not=b_{3}$. We call $N_{2}(\omega,b)$ trivial if
$(b_{2}-b_{3})\sin\theta>0$, $N_{2}(\omega,b)$ non-trivial if
$(b_{2}-b_{3})\sin\theta<0$.

Splitting numbers possess the following properties:

Lemma 3.3. (cf. [Lon1], Lemma 9.1.5 and List 9.1.12 of [Lon2]) For $M\in{\rm
Sp}(2n)$ and ${\omega}\in\mathbb{U}$, there hold

$\displaystyle S_{M}^{\pm}({\omega})$ $\displaystyle=$ $\displaystyle
0,\qquad{\it if}\;\;{\omega}\not\in{\sigma}(M).$ (3.24) $\displaystyle
S_{N_{1}(1,a)}^{+}(1)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1,&\quad{\rm if}\;\;a\geq 0,\cr
0,&\quad{\rm if}\;\;a<0.\cr\end{array}\right.$ (3.27)

For any $M_{i}\in{\rm Sp}(2n_{i})$ with $i=0$ and $1$, there holds

$S^{\pm}_{M_{0}\diamond
M_{1}}({\omega})=S^{\pm}_{M_{0}}({\omega})+S^{\pm}_{M_{1}}({\omega}),\qquad\forall\;{\omega}\in\mathbb{U}.$
(3.28)

The following is the precise index iteration formulae for symplectic paths,
which is due to Long (cf. Theorem 8.3.1 and Corollary 8.3.2 of [Lon2]).

Theorem 3.4. Let $\gamma\in{\mathcal{P}}_{\tau}(2n)$, then there exists a path
$f\in C([0,1],\Omega^{0}(\gamma(\tau))$ such that $f(0)=\gamma(\tau)$ and

$\displaystyle f(1)=$ $\displaystyle N_{1}(1,1)^{\diamond p_{-}}\diamond
I_{2p_{0}}\diamond N_{1}(1,-1)^{\diamond p_{+}}\diamond N_{1}(-1,1)^{\diamond
q_{-}}\diamond(-I_{2q_{0}})\diamond N_{1}(-1,-1)^{\diamond q_{+}}$ (3.29)
$\displaystyle\diamond R(\theta_{1})\diamond\cdots\diamond
R(\theta_{r})\diamond N_{2}(\omega_{1},u_{1})\diamond\cdots\diamond
N_{2}(\omega_{r_{*}},u_{r_{*}})$ $\displaystyle\diamond
N_{2}({\lambda}_{1},v_{1})\diamond\cdots\diamond
N_{2}({\lambda}_{r_{0}},v_{r_{0}})\diamond M_{0}$

where $N_{2}(\omega_{j},u_{j})$s are non-trivial and
$N_{2}({\lambda}_{j},v_{j})$s are trivial basic normal forms;
$\sigma(M_{0})\cap U=\emptyset$; $p_{-}$, $p_{0}$, $p_{+}$, $q_{-}$, $q_{0}$,
$q_{+}$, $r$, $r_{*}$ and $r_{0}$ are non-negative integers;
$\omega_{j}=e^{\sqrt{-1}\alpha_{j}}$, $\lambda_{j}=e^{\sqrt{-1}\beta_{j}}$;
$\theta_{j}$, $\alpha_{j}$, $\beta_{j}$ $\in(0,\pi)\cup(\pi,2\pi)$; these
integers and real numbers are uniquely determined by $\gamma(\tau)$. Then
using the functions defined in (1.3)

$\displaystyle i(\gamma,m)$ $\displaystyle=$ $\displaystyle
m(i(\gamma,1)+p_{-}+p_{0}-r)+2\sum_{j=1}^{r}E\left(\frac{m\theta_{j}}{2\pi}\right)-r-p_{-}-p_{0}$
(3.30)
$\displaystyle-\frac{1+(-1)^{m}}{2}(q_{0}+q_{+})+2\left(\sum_{j=1}^{r_{*}}\varphi\left(\frac{m\alpha_{j}}{2\pi}\right)-r_{*}\right).$
$\displaystyle\nu(\gamma,m)$ $\displaystyle=$
$\displaystyle\nu(\gamma,1)+\frac{1+(-1)^{m}}{2}(q_{-}+2q_{0}+q_{+})+2(r+r_{*}+r_{0})$
(3.31)
$\displaystyle-2\left(\sum_{j=1}^{r}\varphi\left(\frac{m\theta_{j}}{2\pi}\right)+\sum_{j=1}^{r_{*}}\varphi\left(\frac{m\alpha_{j}}{2\pi}\right)+\sum_{j=1}^{r_{0}}\varphi\left(\frac{m\beta_{j}}{2\pi}\right)\right)$
$\displaystyle\hat{i}(\gamma,1)$ $\displaystyle=$ $\displaystyle
i(\gamma,1)+p_{-}+p_{0}-r+\sum_{j=1}^{r}\frac{\theta_{j}}{\pi}.$ (3.32)

In addition, $i(\gamma,1)$ is odd if $f(1)=N_{1}(1,1)$, $I_{2}$,
$N_{1}(-1,1)$, $-I_{2}$, $N_{1}(-1,-1)$ and $R(\theta)$; $i(\gamma,1)$ is even
if $f(1)=N_{1}(1,-1)$ and $N_{2}(\omega,b)$; $i(\gamma,1)$ can be any integer
if $\sigma(f(1))\cap\mathbb{U}=\emptyset$.

We have the following properties in the index iteration theory.

Theorem 3.5.  (cf. Theorem 2.2 of [LoZ]) Let
$\gamma\in{\mathcal{P}}_{\tau}(2n)$ and $M=\gamma(\tau)$. Then for any
$m\in\mathbb{N}$, there holds

$\nu(\gamma,m)-\frac{e(M)}{2}\leq
i(\gamma,m+1)-i(\gamma,m)-i(\gamma,1)\leq\nu(\gamma,1)-\nu(\gamma,m+1)+\frac{e(M)}{2},$

where $e(M)$ is the elliptic hight defined in Section 1.

The common index jump theorem (cf. Theorem 4.3 of [LoZ]) for symplectic paths
has become one of the main tools in studying periodic orbits in Hamiltonian
and symplectic dynamics. In 2016, the following enhanced common index jump
theorem has been obtained by Duan, Long and Wang in [DLW2].

Theorem 3.6. (cf. Theorem 3.5 of [DLW2]) Let
$\gamma_{k}\in\mathcal{P}_{\tau_{k}}(2n)$ for $k=1,\cdots,q$ be a finite
collection of symplectic paths. Let $M_{k}={\gamma}_{k}(\tau_{k})$. We extend
${\gamma}_{k}$ to $[0,+\infty)$ by (3.6) inductively. Suppose

$\hat{i}({\gamma}_{k},1)>0,\qquad\forall\ k=1,\cdots,q.$ (3.33)

Then for any fixed integer $\bar{m}\in\mathbb{N}$, there exist infinitely many
$(q+1)$-tuples $(N,m_{1},\cdots,\\\ m_{q})\in\mathbb{N}^{q+1}$ such that for
all $1\leq k\leq q$ and $1\leq m\leq\bar{m}$, there holds

$\displaystyle\nu({\gamma}_{k},2m_{k}-m)$ $\displaystyle=$
$\displaystyle\nu({\gamma}_{k},2m_{k}+m)=\nu({\gamma}_{k},m),$ (3.34)
$\displaystyle i({\gamma}_{k},2m_{k}+m)$ $\displaystyle=$ $\displaystyle
2N+i({\gamma}_{k},m),$ (3.35) $\displaystyle i({\gamma}_{k},2m_{k}-m)$
$\displaystyle=$ $\displaystyle
2N-i({\gamma}_{k},m)-2(S^{+}_{M_{k}}(1)+Q_{k}(m)),$ (3.36) $\displaystyle
i({\gamma}_{k},2m_{k})$ $\displaystyle=$ $\displaystyle
2N-(S^{+}_{M_{k}}(1)+C(M_{k})-2\Delta_{k}),$ (3.37)

where $S_{M_{k}}^{\pm}({\omega})$ is the splitting number of $M_{k}$ at
${\omega}$ in Definition 3.2 and

$\displaystyle
C(M_{k})=\sum\limits_{0<\theta<2\pi}S^{-}_{M_{k}}(e^{\sqrt{-1}\theta}),\
\Delta_{k}=\sum_{0<\\{m_{k}{\theta}/\pi\\}<\delta}S^{-}_{M_{k}}(e^{\sqrt{-1}{\theta}}),$
$\displaystyle
Q_{k}(m)=\sum_{\theta\in(0,2\pi),e^{\sqrt{-1}{\theta}}\in{\sigma}(M_{k}),\atop\\{\frac{m_{k}{\theta}}{\pi}\\}=\\{\frac{m{\theta}}{2\pi}\\}=0}S^{-}_{M_{k}}(e^{\sqrt{-1}{\theta}}).$
(3.38)

More precisely, by (4.10), (4.40) and (4.41) in [LoZ], we have

$\displaystyle
m_{k}=\left(\left[\frac{N}{\bar{M}\hat{i}(\gamma_{k},1)}\right]+\chi_{k}\right)\bar{M},\quad
1\leq k\leq q,$ (3.39)

where $\chi_{k}=0$ or $1$ for $1\leq k\leq q$ and
$\frac{\bar{M}\theta}{\pi}\in\mathbb{Z}$ whenever
$e^{\sqrt{-1}\theta}\in\sigma(M_{k})$ and $\frac{\theta}{\pi}\in\mathbb{Q}$
for some $1\leq k\leq q$. Furthermore, for any fixed $M_{0}\in\mathbb{N}$, we
may further require $M_{0}|N$, and for any $\epsilon>0$, we can choose $N$ and
$\\{\chi_{k}\\}_{1\leq k\leq q}$ such that

$\displaystyle\left|\left\\{\frac{N}{\bar{M}\hat{i}(\gamma_{k},1)}\right\\}-\chi_{k}\right|<\epsilon,\quad
1\leq k\leq q.$ (3.40)

##  4\. Proof of main results

###  4.1. Proof of Theorem 1.1

In this subsection, we make the following assumption and give the proof of
Theorem 1.1.

(F1) For any Finsler sphere $(S^{n},F)$ $(n\geq 4)$ satisfying
$\left(\frac{2n-3}{n-1}\right)^{2}\left(\frac{{\lambda}}{1+{\lambda}}\right)^{2}<K\leq
1$ and ${\lambda}<\frac{n-1}{n-2}$, there holds
${\mathcal{N}}(S^{n},F)<+\infty$, denoted by $\\{c_{j}\\}_{j=1}^{p}$.

For any $1\leq j\leq p$, we rewrite (3.29) as follows

$\displaystyle f_{j}(1)$ $\displaystyle=$ $\displaystyle
N_{1}(1,1)^{{\diamond}p_{j,-}}\,{\diamond}\,I_{2p_{j,0}}\,{\diamond}\,N_{1}(1,-1)^{{\diamond}p_{j,+}}$
(4.1)
$\displaystyle{\diamond}\,N_{1}(-1,1)^{{\diamond}q_{j,-}}\,{\diamond}\,(-I_{2q_{j,0}})\,{\diamond}\,N_{1}(-1,-1)^{{\diamond}q_{j,+}}$
$\displaystyle{\diamond}\,R({\theta}_{j,1})\,{\diamond}\,\cdots\,{\diamond}\,R({\theta}_{j,r_{j,1}})\,{\diamond}\,R(\tilde{{\theta}}_{j,1})\,{\diamond}\,\cdots\,{\diamond}\,R(\tilde{{\theta}}_{j,r_{j,2}})$
$\displaystyle{\diamond}\,N_{2}(e^{\sqrt{-1}{\alpha}_{j,1}},A_{j,1})\,{\diamond}\,\cdots\,{\diamond}\,N_{2}(e^{\sqrt{-1}{\alpha}_{j,r_{j,3}}},A_{j,r_{j,3}})$
$\displaystyle{\diamond}\,N_{2}(e^{\sqrt{-1}\tilde{{\alpha}}_{j,1}},\tilde{A}_{j,1})\,{\diamond}\,\cdots\,{\diamond}\,N_{2}(e^{\sqrt{-1}\tilde{{\alpha}}_{j,r_{j,4}}},\tilde{A}_{j,r_{j,4}})$
$\displaystyle{\diamond}\,N_{2}(e^{\sqrt{-1}{\beta}_{j,1}},B_{j,1})\,{\diamond}\,\cdots\,{\diamond}\,N_{2}(e^{\sqrt{-1}{\beta}_{j,r_{j,5}}},B_{j,r_{j,5}})$
$\displaystyle{\diamond}\,N_{2}(e^{\sqrt{-1}\tilde{{\beta}}_{j,1}},\tilde{B}_{j,1})\,{\diamond}\,\cdots\,{\diamond}\,N_{2}(e^{\sqrt{-1}\tilde{{\beta}}_{j,r_{j,6}}},\tilde{B}_{j,r_{j,6}})$
$\displaystyle{\diamond}\,H(2)^{{\diamond}h_{j,+}}{\diamond}\,H(-2)^{{\diamond}h_{j,-}},$

where
$\frac{{\theta}_{j,k}}{2\pi}\in\mathbb{Q}\cap(0,1)\setminus\\{\frac{1}{2}\\}$
for $1\leq k\leq r_{j,1}$,
$\frac{\tilde{{\theta}}_{j,k}}{2\pi}\in(0,1)\setminus\mathbb{Q}$ for $1\leq
k\leq r_{j,2}$,
$\frac{{\alpha}_{j,k}}{2\pi}\in\mathbb{Q}\cap(0,1)\setminus\\{\frac{1}{2}\\}$
for $1\leq k\leq r_{j,3}$,
$\frac{\tilde{{\alpha}}_{j,k}}{2\pi}\in(0,1)\setminus\mathbb{Q}$ for $1\leq
k\leq r_{j,4}$,
$\frac{{\beta}_{j,k}}{2\pi}\in\mathbb{Q}\cap(0,1)\setminus\\{\frac{1}{2}\\}$
for $1\leq k\leq r_{j,5}$,
$\frac{\tilde{{\beta}}_{j,k}}{2\pi}\in(0,1)\setminus\mathbb{Q}$ for $1\leq
k\leq r_{j,6}$; $N_{2}(e^{\sqrt{-1}{\alpha}_{j,k}},A_{j,k})$’s and
$N_{2}(e^{\sqrt{-1}\tilde{{\alpha}}_{j,k}},\tilde{A}_{j,k})$’s are nontrivial
and $N_{2}(e^{\sqrt{-1}{\beta}_{j,k}},B_{j,k})$’s and
$N_{2}(e^{\sqrt{-1}\tilde{{\beta}}_{j,k}},\tilde{B}_{j,k})$’s are trivial, and
all non-negative integers satisfy the equality

$\displaystyle
p_{j,-}+p_{j,0}+p_{j,+}+q_{j,-}+q_{j,0}+q_{j,+}+r_{j,1}+r_{j,2}+2\sum_{k=3}^{6}r_{j,k}+h_{j}=n-1.$
(4.2)

Lemma 4.1. Under the assumption (F1), for any prime closed geodesic $c_{j}$,
$1\leq j\leq p$, there holds

$i(c_{j}^{m})\geq\left[\frac{(2n-3)m}{n-1}\right](n-1),\qquad\forall\
m\in\mathbb{N},$ (4.3)

and the average index satisfies

$\hat{i}(c_{j})>2n-3.$ (4.4)

Proof. By the assumption (F1), since the flag curvature $K$ satisfies
$\left(\frac{2n-3}{n-1}\right)^{2}\left(\frac{{\lambda}}{1+{\lambda}}\right)^{2}<K\leq
1$, we can choose
$\left(\frac{2n-3}{n-1}\right)^{2}\left(\frac{\lambda}{\lambda+1}\right)^{2}<\delta\leq
K\leq 1$. Then by Lemma 2 in [Rad5], it yields

$\hat{i}(c_{j})\geq(n-1)\sqrt{\delta}\frac{1+\lambda}{\lambda}>2n-3.$

Note that the length of $c_{j}^{m}$ satisfies $L(c_{j}^{m})=mL(c_{j})\geq
m\pi\frac{1+\lambda}{\lambda}>\frac{(2n-3)m}{n-1}\pi/\sqrt{\delta}$ for $m\geq
1$ and $1\leq j\leq p$ by Theorem 1 of [Rad4]. Then it follows from Lemma 1 of
[Rad5] that $i(c_{j}^{m})\geq[\frac{(2n-3)m}{n-1}](n-1)$.

Combining Lemma 4.1 with Theorem 3.5, it follows that

$\displaystyle i(c_{j}^{m+1})-i(c_{j}^{m})-\nu(c_{j}^{m})\geq
i(c_{j})-\frac{e(P_{c_{j}})}{2}\geq 0,\quad\forall\ m\in\mathbb{N},\ 1\leq
j\leq p.$ (4.5)

Here the last inequality holds by the fact that $e(P_{c_{j}})\leq 2(n-1)$ and
$i(c_{j})\geq n-1$.

It follows from (4.4), Theorem 4.3 in [LoZ] and Theorem 3.6 that for any fixed
integer $\bar{m}\in\mathbb{N}$, there exist infinitely many $(p+1)$-tuples
$(N,m_{1},\cdots,m_{p})\in\mathbb{N}^{p+1}$ such that for any $1\leq j\leq p$
and $1\leq m\leq\bar{m}$, there holds

$\displaystyle i(c_{j}^{2m_{j}-m})+\nu(c_{j}^{2m_{j}-m})$ $\displaystyle=$
$\displaystyle
2N-i(c_{j}^{m})-\left(2S^{+}_{P_{c_{j}}}(1)+2Q_{j}(m)-\nu(c_{j}^{m})\right),$
(4.6) $\displaystyle i(c_{j}^{2m_{j}})$ $\displaystyle\geq$ $\displaystyle
2N-\frac{e(P_{c_{j}})}{2},$ (4.7) $\displaystyle
i(c_{j}^{2m_{j}})+\nu(c_{j}^{2m_{j}})$ $\displaystyle\leq$ $\displaystyle
2N+\frac{e(P_{c_{j}})}{2},$ (4.8) $\displaystyle i(c_{j}^{2m_{j}+m})$
$\displaystyle=$ $\displaystyle 2N+i(c_{j}^{m}),$ (4.9)

where (4.7) and (4.8) follow from (4.32) and (4.33) in Theorem 4.3 in [LoZ]
respectively.

Note that by the List 9.1.12 on page 198 of [Lon2], (3.31), (3.38), (4.1) and
$\nu(c_{j})=p_{j,-}+2p_{j,0}+p_{j,+}$, we have

$\displaystyle S^{+}_{P_{c_{j}}}(1)=p_{j,-}+p_{j,0},$ (4.10) $\displaystyle
C(P_{c_{j}})=q_{j,0}+q_{j,+}+r_{j,1}+r_{j,2}+2r_{j,3}+2r_{j,4},$ (4.11)
$\displaystyle
Q_{j}(m)=\frac{1+(-1)^{m}}{2}(q_{j,0}+q_{j,+})+(r_{j,1}+r_{j,3})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-\sum_{k=1}^{r_{j,1}}{\varphi}\left(\frac{m\theta_{j,k}}{2\pi}\right)-\sum_{k=1}^{r_{j,3}}{\varphi}\left(\frac{m{\alpha}_{j,k}}{2\pi}\right),$
(4.12)
$\displaystyle\nu(c_{j}^{m})=(p_{j,-}+2p_{j,0}+p_{j,+})+\frac{1+(-1)^{m}}{2}(q_{j,-}+2q_{j,0}+q_{j,+})+2(r_{j,1}+r_{j,3}+r_{j,5})$
$\displaystyle\qquad\qquad-2\left(\sum_{k=1}^{r_{j,1}}{\varphi}\left(\frac{m{\theta}_{j,k}}{2\pi}\right)+\sum_{k=1}^{r_{j,3}}{\varphi}\left(\frac{m{\alpha}_{j,k}}{2\pi}\right)+\sum_{k=1}^{r_{j,5}}{\varphi}\left(\frac{m{\beta}_{j,k}}{2\pi}\right)\right).$
(4.13)

By (4.10), (4.12) and (4.13), we obtain

$\displaystyle 2S^{+}_{P_{c_{j}}}(1)+2Q_{j}(m)-\nu(c_{j}^{m})$
$\displaystyle\qquad=p_{j,-}-p_{j,+}-\frac{1+(-1)^{m}}{2}(q_{j,-}-q_{j,+})-2r_{j,5}+2\sum_{k=1}^{r_{j,5}}{\varphi}\left(\frac{m{\beta}_{j,k}}{2\pi}\right),$

which, together with (4.6), gives

$\displaystyle i(c_{j}^{2m_{j}-m})+\nu(c_{j}^{2m_{j}-m})$ $\displaystyle=$
$\displaystyle
2N-i(c_{j}^{m})-p_{j,-}+p_{j,+}+\frac{1+(-1)^{m}}{2}(q_{j,-}-q_{j,+})$ (4.14)
$\displaystyle\quad+2r_{j,5}-2\sum_{k=1}^{r_{j,5}}{\varphi}\left(\frac{m{\beta}_{j,k}}{2\pi}\right),\quad\forall\
1\leq m\leq\bar{m}.$

By (4.7)-(4.9), (4.14), (4.2), (4.3) and the fact $e(P_{c_{j}})\leq 2(n-1)$,
there holds

$\displaystyle i(c_{j}^{2m_{j}-m})+\nu(c_{j}^{2m_{j}-m})$ $\displaystyle\leq$
$\displaystyle 2N+n-1-\left[\frac{(2n-3)m}{n-1}\right](n-1),$ (4.15)
$\displaystyle\qquad\qquad\qquad\qquad\forall\ 1\leq m\leq\bar{m},$
$\displaystyle 2N-(n-1)$ $\displaystyle\leq$ $\displaystyle
2N-\frac{e(P_{c_{j}})}{2}\leq i(c_{j}^{2m_{j}}),$ (4.16) $\displaystyle
i(c_{j}^{2m_{j}})+\nu(c_{j}^{2m_{j}})$ $\displaystyle\leq$ $\displaystyle
2N+\frac{e(P_{c_{j}})}{2}\leq 2N+n-1,$ (4.17) $\displaystyle
2N+\left[\frac{(2n-3)m}{n-1}\right](n-1)$ $\displaystyle\leq$ $\displaystyle
2N+i(c_{j}^{m})=i(c_{j}^{2m_{j}+m}),\ \forall\ 1\leq m\leq\bar{m}.$ (4.18)

Note that by (4.5), we have

$\displaystyle i(c_{j}^{m})\leq i(c_{j}^{m+1}),\qquad
i(c_{j}^{m})+\nu(c_{j}^{m})\leq i(c_{j}^{m+1})+\nu(c_{j}^{m+1}),\quad\forall
m\in\mathbb{N},$

which, together with (4.15) and (4.18), implies

$\displaystyle i(c_{j}^{m})+\nu(c_{j}^{m})\leq
i(c_{j}^{2m_{j}-\bar{m}})+\nu(c_{j}^{2m_{j}-\bar{m}})$
$\displaystyle\qquad\leq
2N+n-1-\left[\frac{(2n-3)\bar{m}}{n-1}\right](n-1),\,\forall\ 1\leq m\leq
2m_{j}-\bar{m},$ (4.19) $\displaystyle
2N+\left[\frac{(2n-3)\bar{m}}{n-1}\right](n-1)\leq
i(c_{j}^{2m_{j}+\bar{m}})\leq i(c_{j}^{m}),\,\forall\ m\geq 2m_{j}+\bar{m}.$
(4.20)

Fix $\bar{m}\geq 3$, next we will obtain more accurate index estimates than
(4.15) about $c_{j}^{2m_{j}-m}$ for $j\in\\{1,\cdots,p\\}$ and $m=1,2$.

Lemma 4.2. Under the assumption (F1), for $j=1,\cdots,p$, we have

$\displaystyle i(c_{j}^{2m_{j}-1})+\nu(c_{j}^{2m_{j}-1})$ $\displaystyle\leq$
$\displaystyle 2N-(n-1),$ (4.21) $\displaystyle
i(c_{j}^{2m_{j}-2})+\nu(c_{j}^{2m_{j}-2})$ $\displaystyle\leq$ $\displaystyle
2N-3(n-1).$ (4.22)

Proof. We give the proofs of (4.21) and (4.22) respectively.

Step 1. The proof of (4.21).

By (3.32) and (4.1), we have

$\displaystyle\hat{i}(c_{j})$ $\displaystyle=$ $\displaystyle
i(c_{j})+p_{j,-}+p_{j,0}-r_{j,1}-r_{j,2}+\sum_{k=1}^{r_{j,1}}\frac{\theta_{j,k}}{\pi}+\sum_{k=1}^{r_{j,2}}\frac{\tilde{\theta}_{j,k}}{\pi}$
(4.23) $\displaystyle\leq$ $\displaystyle
i(c_{j})+p_{j,-}+p_{j,0}+r_{j,1}+r_{j,2}.$

Combining this with (4.4), we get

$\displaystyle i(c_{j})+p_{j,-}+p_{j,0}+r_{j,1}+r_{j,2}\geq 2n-2.$ (4.24)

Then, using (4.14), (4.24) and (4.2), we obtain

$\displaystyle i(c_{j}^{2m_{j}-1})+\nu(c_{j}^{2m_{j}-1})$ $\displaystyle=$
$\displaystyle 2N-i(c_{j})-p_{j,-}+p_{j,+}$ (4.25) $\displaystyle\leq$
$\displaystyle 2N-(2n-2)+p_{j,0}+r_{j,1}+r_{j,2}+p_{j,+}$ $\displaystyle\leq$
$\displaystyle 2N-(n-1),$

which prove the inequality (4.21).

Step 2. The proof of (4.22).

By (4.14), we have

$\displaystyle
i(c_{j}^{2m_{j}-2})+\nu(c_{j}^{2m_{j}-2})=2N-i(c_{j}^{2})-p_{j,-}+p_{j,+}+q_{j,-}-q_{j,+}.$
(4.26)

Using (4.1), (3.30), (3.32) and the functions in (1.3), we obtain

$\displaystyle i(c_{j}^{2})$ $\displaystyle=$ $\displaystyle
2(i(c_{j})+p_{j,-}+p_{j,0}-r_{j,1}-r_{j,2})+2\sum_{k=1}^{r_{j,1}}E\left(\frac{\theta_{j,k}}{\pi}\right)+2\sum_{k=1}^{r_{j,2}}E\left(\frac{\tilde{\theta}_{j,k}}{\pi}\right)$
(4.27) $\displaystyle-r_{j,1}-r_{j,2}-p_{j,-}-p_{j,0}-q_{j,0}-q_{j,+},$
$\displaystyle\hat{i}(c_{j})$ $\displaystyle=$ $\displaystyle
i(c_{j})+p_{j,-}+p_{j,0}-r_{j,1}-r_{j,2}+\sum_{k=1}^{r_{j,1}}\frac{\theta_{j,k}}{\pi}+\sum_{k=1}^{r_{j,2}}\frac{\tilde{\theta}_{j,k}}{\pi}.$
(4.28)

Combining these equalities (4.27) and (4.28), we get

$\displaystyle i(c_{j}^{2})$ $\displaystyle=$ $\displaystyle
2\hat{i}(c_{j})-2\sum_{k=1}^{r_{j,1}}\left\\{\frac{\theta_{j,k}}{\pi}\right\\}-2\sum_{k=1}^{r_{j,2}}\left\\{\frac{\tilde{\theta}_{j,k}}{\pi}\right\\}$
(4.29) $\displaystyle+r_{j,1}+r_{j,2}-p_{j,-}-p_{j,0}-q_{j,0}-q_{j,+}.$

Note that $\sum_{k=1}^{r_{j,1}}\left\\{\frac{\theta_{j,k}}{\pi}\right\\}\leq
r_{j,1}$ and
$\sum_{k=1}^{r_{j,2}}\left\\{\frac{\tilde{\theta}_{j,k}}{\pi}\right\\}\leq
r_{j,2}$. Combining this with (4.26), (4.29), (4.4) and (4.2), we conclude

$\displaystyle i(c_{j}^{2m_{j}-2})+\nu(c_{j}^{2m_{j}-2})$ $\displaystyle<$
$\displaystyle 2N-(4n-6)+r_{j,1}+r_{j,2}+p_{j,+}+p_{j,0}+q_{j,0}+q_{j,-}$
(4.30) $\displaystyle\leq$ $\displaystyle 2N-(4n-6)+n-1=2N-(3n-5),$

which implies that

$\displaystyle i(c_{j}^{2m_{j}-2})+\nu(c_{j}^{2m_{j}-2})\leq 2N-(3n-4).$
(4.31)

Next, assuming that $i(c_{j}^{2m_{j}-2})+\nu(c_{j}^{2m_{j}-2})=2N-(3n-4)$,
then by (4.26) and (4.30) we have

$\displaystyle i(c_{j}^{2})+p_{j,-}-p_{j,+}-q_{j,-}+q_{j,+}$ $\displaystyle=$
$\displaystyle 3n-4,$ (4.32) $\displaystyle
r_{j,1}+r_{j,2}+p_{j,+}+p_{j,0}+q_{j,0}+q_{j,-}$ $\displaystyle=$
$\displaystyle n-1.$ (4.33)

Combining (4.32), (4.33) and (4.2), it yields

$\displaystyle p_{j,-}+q_{j,+}$ $\displaystyle=$ $\displaystyle 0,$ (4.34)
$\displaystyle i(c_{j}^{2})-p_{j,+}-q_{j,-}$ $\displaystyle=$ $\displaystyle
3n-4.$ (4.35)

However, from (4.27), (4.33) and (4.34), we deduce

$\displaystyle i(c_{j}^{2})-p_{j,+}-q_{j,-}$ $\displaystyle=$ $\displaystyle
r_{j,1}+r_{j,2}+p_{j,-}+p_{j,0}+q_{j,0}+q_{j,+}+p_{j,+}+q_{j,-}\quad({\rm
mod}\,2)$ (4.36) $\displaystyle=$ $\displaystyle
r_{j,1}+r_{j,2}+p_{j,0}+q_{j,0}+p_{j,+}+q_{j,-}\quad({\rm mod}\,2)$
$\displaystyle=$ $\displaystyle n-1\quad({\rm mod}\,2),$

which contradicts (4.35). Thus $i(c_{j}^{2m_{j}-2})+\nu(c_{j}^{2m_{j}-2})\leq
2N-(3n-4)-1=2N-3(n-1)$.

Now Steps 1-2 complete the proof of Lemma 4.2.

The following lemma gives the existence of one elliptic closed geodesic
satisfying some special properties on certain positively curved $(S^{n},F)$.

Lemma 4.3. (cf. Lemma 3.1 of [Dua1] and Section 3 of [Dua2]) Under the
assumption $(\frac{{\lambda}}{1+{\lambda}})^{2}<K\leq 1$, there exist at least
two prime elliptic closed geodesics on $(S^{n},F)$, and one of them, saying
$c_{j_{0}}$, satisfying the following properties

$\displaystyle
i(c_{j_{0}}^{2m_{j_{0}}})+\nu(c_{j_{0}}^{2m_{j_{0}}})=2N+n-1,\quad\overline{C}_{2N+n-1}(E,c_{j_{0}}^{2m_{j_{0}}})=\mathbb{Q},$
(4.37) $\displaystyle
p_{{j_{0}},-}=q_{{j_{0}},+}=r_{{j_{0}},3}=r_{{j_{0}},4}=r_{{j_{0}},6}=h_{j_{0}}=0,$
(4.38) $\displaystyle r_{j_{0},2}=\Delta_{1}\geq 1,$ (4.39)

where the freedom of choose and existence of $N$ and $m_{j_{0}}$ are specified
in Theorem 3.6, and all integers in (4.38) and (4.39) are defined in Theorem
3.4 and Theorem 3.6.

Proof of Theorem 1.1. We use two following steps to prove Theorem 1.1.

Step 1. The existence of $n-1$ prime closed geodesics.

By Lemma 2.6, for every $i\in\mathbb{N}$, there exist some
$m(i),j(i)\in\mathbb{N}$ such that

$\displaystyle
E(c_{j(i)}^{m(i)})=\kappa_{i},\quad\overline{C}_{2i+\dim(z)-2}(E,c_{j(i)}^{m(i)})\neq
0,$ (4.40)

where $\kappa_{i}\neq\kappa_{i^{\prime}}$ for $i\neq i^{\prime}$ by Lemma 2.5.
It follows from Subsection 2.2 that $\dim(z)=n+1$. By combining Proposition
2.1 with (4.40), we have

$\displaystyle i(c_{j(i)}^{m(i)})\leq 2i+n-1\leq
i(c_{j(i)}^{m(i)})+\nu(c_{j(i)}^{m(i)}),\qquad\forall\ i\in\mathbb{N}.$ (4.41)

Note that for any $i\in G_{1}\equiv[N-(n-2),N-1]\cap\mathbb{N}$, there holds
$2i+n-1\in\bar{G_{1}}\equiv[2N-(n-3),2N+(n-3)]\cap\mathbb{N}$.

Claim 1. The following fact holds

$\displaystyle m(i)=2m_{j(i)},\quad\forall\ i\in G_{1}.$ (4.42)

In fact, assuming that $m(i_{0})\neq 2m_{j(i_{0})}$ for some $i_{0}\in G_{1}$.
If $m(i_{0})=2m_{j(i_{0})}-m$ with $m\in\mathbb{N}$, then by (4.15), (4.19),
(4.21) and $2i_{0}+n-1\in\bar{G_{1}}$, we have
$i(c_{j(i_{0})}^{m(i_{0})})+\nu(c_{j(i_{0})}^{m(i_{0})})\leq
2N-(n-1)<2i_{0}+n-1$, which contradicts (4.41). If $m(i_{0})=2m_{j(i_{0})}+m$
with $m\in\mathbb{N}$, then by (4.18), (4.20) and $2i_{0}+n-1\in\bar{G_{1}}$,
we have $i(c_{j(i_{0})}^{m(i_{0})})\geq 2N+(n-1)>2i_{0}+n-1$, which again
contradicts (4.41). This proves Claim 1.

Consequently it yields $c_{j(i)}\neq c_{j(i^{\prime})}$ for any two different
$i,i^{\prime}\in G_{1}$ since
$E(c_{j(i)}^{2m_{j(i)}})=\kappa_{i}\neq\kappa_{i^{\prime}}=E(c_{j(i^{\prime})}^{2m_{j(i^{\prime})}})$.
Thus there exist at least $n-2$ prime closed geodesics on $(S^{n},F)$, which
is denoted by $\\{c_{1},\ldots,c_{n-2}\\}$ after a permutation of
$\\{1,\ldots,p\\}$, i.e.

$\displaystyle\\{j(i)\mid i\in G_{1}\\}=\\{1,2,\cdots,n-2\\}.$ (4.43)

By Lemma 4.3, there exists $j_{0}\in\\{1,\dots,p\\}$ such that
$\overline{C}_{2N+n-1}(E,c_{j_{0}}^{2m_{j_{0}}})\neq 0$. So, by Proposition
2.2 and (4.17), we have

$\displaystyle\overline{C}_{q}(E,c_{j_{0}}^{2m_{j_{0}}})=0,\quad\forall q\neq
2N+(n-1).$ (4.44)

From (4.40) and (4.44), we conclude that
$c_{j_{0}}\notin\\{c_{1},\ldots,c_{n-2}\\}$. Hence there exist at least $n-1$
closed geodesics on $(S^{n},F)$.

Step 2. The existence of another prime closed geodesic.

We carry out this proof by contradiction. Suppose there exist only $n-1$
closed geodesics on $(S^{n},F)$, i.e. $p=n-1$. And, as inferred in Step 1, we
have $c_{j_{0}}=c_{n-1}$.

In order to get a contradiction, we introduce
$G_{2}\equiv[N-(2n-3),N-(n-1)]\cap\mathbb{N}$ and
$\bar{G_{2}}\equiv[2N-3(n-1)+2,2N-(n-1)]\cap\mathbb{N}$. Note that $i\in
G_{2}$ implies that $2i+n-1\in\bar{G_{2}}$, and $G_{1}\cap
G_{2}=\bar{G_{1}}\cap\bar{G_{2}}=\emptyset$. By (4.15)-(4.20), (4.41), Lemma
4.3 and Proposition 2.2, it follows that
$m(i)=2m_{j(i)}\;\text{or}\;2m_{j(i)}-1$ for any $i\in G_{2}$.

Claim 2. We have the following

$\displaystyle m(i)=2m_{j(i)}-1,\quad\forall\;i\in G_{2}.$ (4.45)

In fact, we assume that $m(i_{0}^{\prime})=2m_{j(i_{0}^{\prime})}$ for some
$i_{0}\in G_{2}$, then $j(i_{0}^{\prime})\neq n-1$ by (4.40) and (4.44). Thus
by Step 1, we must have $j(i_{0}^{\prime})=j(i_{0}^{\prime\prime})$ with some
$i_{0}^{\prime\prime}\in\\{1,2,\cdots,n-2\\}$. Then by (4.40) and (4.42), we
have
$\kappa_{i_{0}^{\prime}}=E(c_{j(i_{0}^{\prime})}^{2m_{j(i_{0}^{\prime})}})=E(c_{j(i_{0}^{\prime\prime})}^{2m_{j(i_{0}^{\prime\prime})}})=\kappa_{i_{0}^{\prime\prime}}$.
This is a contradiction by Lemma 2.5 since ${\mathcal{N}}(S^{n},F)<+\infty$.
So Claim 2 holds.

Consequently it yields $c_{j(i)}\neq c_{j(i^{\prime})}$ for any
$i,i^{\prime}\in G_{2}$ and $i\neq i^{\prime}$ since
$E(c_{j(i)}^{2m_{j(i)}-1})=\kappa_{i}\neq\kappa_{i^{\prime}}=E(c_{j(i^{\prime})}^{2m_{j(i^{\prime})}-1})$.
This implies

$\displaystyle\\{j(i)\mid i\in G_{2}\\}=\\{1,2,\cdots,n-1\\}.$ (4.46)

Claim 3. There holds

$\displaystyle\overline{C}_{2N-(n-1)}(E,c_{j}^{m})\neq 0,\;\text{only
if}\;j=j(N-(n-1))\;\text{and}\;m=2m_{j(N-n+1)}-1.$ (4.47)

In fact, from (4.15)-(4.20), Lemma 4.3 and Proposition 2.1, we can deduce that
for any $j\in\\{1,2,\cdots,n-1\\}$ there holds

$\displaystyle\overline{C}_{2N-(n-1)}(E,c_{j}^{m})\neq 0,\quad\text{only if}\
m\in\\{2m_{j},2m_{j}-1\\}.$ (4.48)

Firstly, assuming that for some $j_{0}^{\prime}\in\\{1,2,\cdots,n-1\\}$, there
holds $\overline{C}_{2N-(n-1)}(E,c_{j_{0}^{\prime}}^{2m_{j_{0}^{\prime}}})\\\
\neq 0$. This, together with (4.16), Propositions 2.1 and 2.2, and (4.44),
implies that

$\displaystyle\overline{C}_{q}(E,c_{j_{0}^{\prime}}^{2m_{j_{0}^{\prime}}})=0\quad\text{for}\;q\neq
2N-(n-1)$ (4.49)

and

$\displaystyle c_{j_{0}^{\prime}}\neq c_{j_{0}},\quad\text{i.e.}\quad
j_{0}^{\prime}\neq n-1.$ (4.50)

By (4.50) and (4.43), we have $j_{0}^{\prime}=j(i_{1})$ for some $i_{1}\in
G_{1}$, and by (4.40) and (4.42), we have
$\overline{C}_{2i_{1}+n-1}(E,c_{j(i_{1})}^{2m_{j(i_{1})}})\neq 0$, i.e., there
holds

$\displaystyle\overline{C}_{2i_{1}+n-1}(E,c_{j_{0}^{\prime}}^{2m_{j_{0}^{\prime}}})\neq
0\quad\text{for some}\;i_{1}\in G_{1}.$ (4.51)

However, note that $2i_{1}+n-1\in\bar{G_{1}}$, $2N-(n-1)\in\bar{G_{2}}$ and
$\bar{G_{1}}\cap\bar{G_{2}}=\emptyset$, so it yields $2i_{1}+n-1\neq
2N-(n-1)$, which implies that (4.51) contradicts (4.49).

Secondly, assuming that for some
$j_{0}^{\prime\prime}\in\\{1,2,\cdots,n-1\\}\setminus\\{j(N-(n-1))\\}$, there
holds
$\overline{C}_{2N-(n-1)}(E,c_{j_{0}^{\prime\prime}}^{2m_{j_{0}^{\prime\prime}}-1})\neq
0,$ which, together with (4.21) and Propositions 2.1 and 2.2, implies that

$\displaystyle\overline{C}_{q}(E,c_{j_{0}^{\prime\prime}}^{2m_{j_{0}^{\prime\prime}}-1})=0\quad\text{for}\;q\neq
2N-(n-1).$ (4.52)

By (4.46), we know that $j_{0}^{\prime\prime}=j(i_{2})$ for some $i_{2}\in
G_{2}\setminus\\{N-(n-1)\\}$, and by (4.40) and (4.45), we have
$\overline{C}_{2i_{2}+n-1}(E,c_{j(i_{2})}^{2m_{j(i_{2})}-1})\neq 0$, then
there holds

$\displaystyle\overline{C}_{2i_{2}+n-1}(E,c_{j_{0}^{\prime\prime}}^{2m_{j_{0}^{\prime\prime}}-1})\neq
0\quad\text{for some}\;i_{2}\in G_{2}\setminus\\{N-(n-1)\\}.$ (4.53)

However, note that $2i_{2}+n-1\in\bar{G_{2}}\setminus\\{2N-(n-1)\\}$, so
(4.53) contradicts (4.52).

The above two contradictions, together with (4.48), complete the proof of
Claim 3.

Combining (4.47), (4.40), (4.45) and Proposition 2.1, we obtain

$\displaystyle
M_{2N-(n-1)}=M_{2N-(n-1)}(\overline{{\Lambda}}S^{n},\overline{{\Lambda}}^{0}S^{n})=\dim\overline{C}_{2N-(n-1)}(E,c_{j(N-n+1)}^{2m_{j(N-n+1)}-1})=1.$
(4.54)

However, by Theorem 3.6, choosing $N$ as a multiple of $n-1$, we obtain
$1=M_{2N-(n-1)}\geq b_{2N-(n-1)}=2$ by Theorems 2.6 and 2.7, which is a
contradiction.

Now Steps 1 and 2 complete the proof of Theorem 1.1.

###  4.2. Proof of Theorem 1.2 and Theorem 1.3

In this subsection, we carry out the proofs of Theorems 1.2 and 1.3 under the
assumption (F1).

Proof of Theorem 1.2. Under the assumption (F1), for those closed geodesics
$\\{c_{j}\\}_{j=1}^{p}$, if $c_{j}$ is hyperbolic, then by (4.16) and (4.17),
there holds

$i(c_{j}^{2m_{j}})=2N=i(c_{j}^{2m_{j}})+\nu(c_{j}^{2m_{j}}).$

Thus for every hyperbolic $c_{j}$, by Proposition 2.1 we have

$\displaystyle\overline{C}_{q}(E,c_{j}^{2m_{j}})=0,\quad\forall\;q\neq 2N.$
(4.55)

By (4.40), (4.42) and (4.55), the closed geodesics
$\\{c_{1},\ldots,c_{n-2}\\}$ found in Step 1 in the proof of Theorem 1.1 are
non-hyperbolic when $n$ is even, and at least $n-3$ of them are non-hyperbolic
when $n$ is odd. By Lemma 4.3, the closed geodesic $c_{j_{0}}$ is non-
hyperbolic. Therefore there are $2[\frac{n}{2}]-1$ non-hyperbolic closed
geodesics on $S^{n}$.

Proof of Theorem 1.3. Here we mainly follow some ideas in the proof of Theorem
1.4 in [Wan3]. For the reader’s conveniences, we give some necessary details.

Claim 4. There are at least $n-2$ closed geodesics $c_{j_{k}}$ for $1\leq
k\leq n-2$ on $(S^{n},F)$ such that
$c_{j_{k}}\in\mathcal{V}_{\infty}(S^{n},F)$ for $1\leq k\leq n-2$.

As in the proof of Theorem 1.1, for any $N$ chosen in (4.6)-(4.9) and
$2i+\dim(z)-2\in(2N-n+1,\,2N+n-1)$, there exist some $1\leq j(i)\leq p$ such
that $c_{j(i)}$ is $(2m_{j(i)},i)$-variationaly visible by (4.40) and (4.42).
Moreover, if $2N-n+1<i_{1}\neq i_{2}<2N+n-1$, then there must hold
$j(i_{1})\neq j(i_{2})$. Hence the map

$\displaystyle\Psi:(2\mathbb{N}+\dim(z)-2)\cap(2N-n+1,2N+n-1)$
$\displaystyle\longrightarrow$ $\displaystyle\\{c_{j}\\}_{1\leq j\leq p},$
$\displaystyle 2i+\dim(z)-2$ $\displaystyle\mapsto$ $\displaystyle c_{j(i)}$
(4.56)

is injective. Here when there are more than one $c_{j}$ satisfying (4.40), we
take any one of them. Since the choose of $N$ satisfying (4.6)-(4.9) is
infinite and ${\mathcal{N}}(S^{n},F)<+\infty$, so Claim 4 holds.

Claim 5. Among the $n-2$ closed geodesics $\\{c_{j_{k}}\\}_{1\leq k\leq n-2}$
found in Claim 4, there are at least $n-3$ ones possessing irrational average
indices.

By Theorem 3.6, we can choose infinitely many $N$ satisfying (4.6)-(4.9) and

$\frac{N}{\bar{M}\hat{i}(c_{j})}\in\mathbb{N}\quad{\rm
and}\quad\chi_{j}=0,\qquad{\rm if}\quad\hat{i}(c_{j})\in\mathbb{Q}.$ (4.57)

Now suppose $\hat{i}(c_{j_{k_{1}}})\in\mathbb{Q}$ and
$\hat{i}(c_{j_{k_{2}}})\in\mathbb{Q}$ hold for some $1\leq k_{1}\neq k_{2}\leq
n-2$. Then by (3.39) and (4.57) it yields

$\displaystyle 2m_{j_{k_{1}}}\hat{i}(c_{j_{k_{1}}})$ $\displaystyle=$
$\displaystyle
2\left(\left[\frac{N}{\bar{M}\hat{i}(c_{j_{k_{1}}})}\right]+\chi_{j_{k_{1}}}\right)\bar{M}\hat{i}(c_{j_{k_{1}}})$
(4.58) $\displaystyle=$ $\displaystyle
2\left(\frac{N}{\bar{M}\hat{i}(c_{j_{k_{1}}})}\right)\bar{M}\hat{i}(c_{j_{k_{1}}})=2N=2\left(\frac{N}{\bar{M}\hat{i}(c_{j_{k_{2}}})}\right)\bar{M}\hat{i}(c_{j_{k_{2}}})$
$\displaystyle=$ $\displaystyle
2\left(\left[\frac{N}{\bar{M}\hat{i}(c_{j_{k_{2}}})}\right]+\chi_{j_{k_{2}}}\right)\bar{M}\hat{i}(c_{j_{k_{2}}})=2m_{j_{k_{2}}}\hat{i}(c_{j_{k_{2}}}).$

In addition, by (4.56) we have

$\Psi(2i_{1}+\dim(z)-2)=c_{j_{k_{1}}},\quad\Psi(2i_{2}+\dim(z)-2)=c_{j_{k_{2}}}\quad{\rm
for\;some}\;i_{1}\neq i_{2}.$ (4.59)

Thus by (4.40), (4.42), Lemma 2.5 and the assumption
${\mathcal{N}}(S^{n},F)<+\infty$, we have

$E(c_{j_{k_{1}}}^{2m_{j_{k_{1}}}})=\kappa_{i_{1}}\neq\kappa_{i_{2}}=E(c_{j_{k_{2}}}^{2m_{j_{k_{2}}}}).$
(4.60)

Since $c_{j_{k_{1}}},c_{j_{k_{2}}}\in\mathcal{V}_{\infty}(S^{n},F)$, by
Theorem 2.8 there holds

$\frac{\hat{i}(c_{j_{k_{1}}})}{L(c_{j_{k_{1}}})}=2\sigma=\frac{\hat{i}(c_{j_{k_{2}}})}{L(c_{j_{k_{2}}})}.$
(4.61)

Note that for any closed geodesic $c$ on $(S^{n},F)$, the following relations
hold

$L(c^{m})=mL(c),\quad\hat{i}(c^{m})=m\hat{i}(c),\quad
L(c)=\sqrt{2E(c)},\qquad\forall m\in\mathbb{N}.$ (4.62)

Hence it follows from (4.60)-(4.62) that

$\displaystyle 2m_{j_{k_{1}}}\hat{i}(c_{j_{k_{1}}})$ $\displaystyle=$
$\displaystyle 2\sigma\cdot
2m_{j_{k_{1}}}L(c_{j_{k_{1}}})=2\sigma\sqrt{2E(c_{j_{k_{1}}}^{2m_{j_{k_{1}}}})}=2\sigma\sqrt{2\kappa_{i_{1}}}$
$\displaystyle\neq$ $\displaystyle
2\sigma\sqrt{2\kappa_{i_{2}}}=2\sigma\sqrt{2E(c_{j_{k_{2}}}^{2m_{j_{k_{2}}}})}=2\sigma\cdot
2m_{j_{k_{2}}}L(c_{j_{k_{2}}})=2m_{j_{k_{2}}}\hat{i}(c_{j_{k_{2}}}),$

which contradicts (4.58), and then there must be
$\hat{i}(c_{j_{k_{1}}})\in\mathbb{R}\setminus\mathbb{Q}$ or
$\hat{i}(c_{j_{k_{2}}})\in\mathbb{R}\setminus\mathbb{Q}$. Hence there is at
most one $1\leq k\leq n-2$ such that $\hat{i}(c_{j_{k}})\in\mathbb{Q}$, i.e.,
there are at least $n-3$ ones possessing irrational average indices. This
proves Claim 5.

Suppose $c_{j_{k}}$ is any closed geodesic found in Claim 5, then we have
$\hat{i}(c_{j_{k}})\in\mathbb{R}\setminus\mathbb{Q}$. Therefore by Theorem
3.4, the linearized Poincaré map $P_{c_{j_{k}}}$ of $c_{j_{k}}$ must contains
a term $R(\theta)$ with $\frac{\theta}{\pi}\notin\mathbb{Q}$. By Lemma 4.3,
the closed geodesic $c_{j_{0}}$ also has this property. Moreover, it follows
from the proof of Theorem 1.1 that $c_{j_{0}}\notin\\{c_{j_{k}}\\}_{1\leq
k\leq n-2}$.

This completes the proof of Theorem 1.3.

###  4.3. Proof of Theorem 1.4

To prove Theorem 1.4, in this subsection we assume the following:

(F2) For any Finsler sphere $(S^{n},F)$ $(n\geq 4)$ satisfying
$\frac{9}{4}\left(\frac{\lambda}{\lambda+1}\right)^{2}<K\leq 1$ and
$\lambda<2$, there holds ${\mathcal{N}}(S^{n},F)<+\infty$, denoted by
$\\{c_{j}\\}_{j=1}^{p}$.

Firstly, we can estimate the index of closed geodesics on $(S^{n},F)$.

Lemma 4.4. Under the assumption (F2), for any prime closed geodesic
$c_{j},1\leq j\leq p$, there holds

$\displaystyle
i\left(c_{j}^{m}\right)\geq\left[\frac{3m}{2}\right](n-1)\qquad\forall
m\in\mathbb{N}.$ (4.63)

Proof. By According to Theorem 1 of [Rad4], we have

$\displaystyle L(c_{j}^{m})=mL(c_{j})\geq m\pi\frac{\lambda+1}{\lambda}.$
(4.64)

By (F2) we can choose
$\delta>\frac{9}{4}\left(\frac{\lambda}{1+\lambda}\right)^{2}$ such that
$K\geq\delta$, it together with (4.64) gives
$L(c_{j}^{m})>\frac{3m\pi}{2\sqrt{\delta}}$, which implies
$i\left(c_{j}^{m}\right)\geq\left[\frac{3m}{2}\right](n-1)$ by Lemma 1 of
[Rad5].

Proof of Theorem 1.4. According to Lemma 4.4, every prime closed geodesic
$c_{j}$, $1\leq j\leq p$, on $(S^{n},F)$ must satisfy

$\displaystyle i(c_{j}^{m})\geq\left[\frac{3m}{2}\right](n-1)\geq
n-1\qquad\forall\ m\geq 1,$ (4.65)

and

$\displaystyle i(c_{j}^{m})\geq\left[\frac{3m}{2}\right](n-1)\geq
3(n-1)\qquad\forall\ m\geq 2.$ (4.66)

As in the proof of Theorem 1.1, for every $i\in\mathbb{N}$, there exist some
$m(i),j(i)\in\mathbb{N}$ such that

$\displaystyle
E(c_{j(i)}^{m(i)})=\kappa_{i},\quad\overline{C}_{2i+n-1}(E,c_{j(i)}^{m(i)})\neq
0,$ (4.67)

where $\kappa_{i}\neq\kappa_{i^{\prime}}$ for $i\neq i^{\prime}$. And then we
have

$\displaystyle i(c_{j(i)}^{m(i)})\leq 2i+n-1\leq
i(c_{j(i)}^{m(i)})+\nu(c_{j(i)}^{m(i)})$ (4.68)

for any $i\in\mathbb{N}$. Thus by (4.65) and (4.66), we obtain

$\displaystyle m(i)=1,\qquad\forall\,\ 1\leq i\leq n-2.$ (4.69)

Consequently it yields $c_{j(i)}\neq c_{j(i^{\prime})}$ for $1\leq i\neq
i^{\prime}\leq n-2$ due to
$E(c_{j(i)})=\kappa_{i}\neq\kappa_{i^{\prime}}=E(c_{j(i^{\prime})})$. Thus
there exist at least $n-2$ prime closed geodesics on $(S^{n},F)$, and we
denote these closed geodesics by $\\{c_{1},\ldots,c_{n-2}\\}$ after a
permutation of $\\{1,\ldots,p\\}$.

According to Theorem 2.3, we have
$b_{n-1}=b_{n-1}(\overline{{\Lambda}}S^{n},\overline{{\Lambda}}^{0}S^{n})=1$.
Then, by Theorem 2.4, we have
$M_{n-1}(\overline{{\Lambda}}S^{n},\overline{{\Lambda}}^{0}S^{n})\geq
b_{n-1}=1$, which, together with (4.66), implies that there exists
$j_{0}\in\\{1,\ldots,p\\}$ such that $\overline{C}_{n-1}(E,c_{j_{0}})\neq 0$.
So, by Proposition 2.2 and (4.65), we have

$\displaystyle\overline{C}_{q}(E,c_{j_{0}})=0,\quad\forall q\neq n-1.$ (4.70)

From (4.67) and (4.70), we conclude that
$c_{j_{0}}\notin\\{c_{1},\ldots,c_{n-2}\\}$. Hence there exist at least $n-1$
closed geodesics on $(S^{n},F)$.

The proof of Theorem 1.4 is complete.

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