In this section, we prove our main theorem by using the mean index equality in Theorem 2.3, the Morse inequality in Theorem 2.5, and the index iteration theory developed by Long and his coworkers, especially a new observation on a symmetric property for closed geodesics in the common index jump intervals, i.e., Lemma 3.2.

First, we make the assumption that

$\text{there exist only finitely many closed geodesics}{c}_{k}\text{,}k=1,\mathrm{\dots},q\text{, on}({S}^{3},F)\text{with reversibility}\lambda $$\text{and flag curvature}K\text{satisfying}(9/4){(\lambda /(1+\lambda ))}^{2}<K\le 1\text{with}\lambda <2\text{.}$

Then, we have an estimate on the index and on the mean index of ${c}_{k}$.

*We have $i\mathit{}\mathrm{(}{c}_{k}\mathrm{)}\mathrm{\ge}\mathrm{2}$ and $\widehat{\u0131}\mathit{}\mathrm{(}{c}_{k}\mathrm{)}\mathrm{>}\mathrm{3}$ for $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}q$.*

#### Proof.

By assumption, since the flag curvature *K* satisfies $(9/4){(\lambda /(1+\lambda ))}^{2}<K\le 1$, we can choose a δ in [31, Lemma 2] to satisfy

$\delta >\frac{9}{4}{\left(\frac{\lambda}{1+\lambda}\right)}^{2}$

and

$\widehat{\u0131}({c}_{k})\ge 2\sqrt{\delta}\frac{\lambda +1}{\lambda}>3.$

The claim $i({c}_{k})\ge 2$ follows from [30, Theorem 3 and Lemma 3].
∎

Combining Lemma 3.1 with Theorem 2.8, it follows that

$i({c}_{k}^{m+1})-i({c}_{k}^{m})-\nu ({c}_{k}^{m})\ge i({c}_{k})-\frac{e({P}_{{c}_{k}})}{2}\ge 0$(3.1)

for all $m\in \mathbb{N}$. Here, the last inequality holds by the fact that $e({P}_{{c}_{k}})\le 4$.

It follows from Lemma 3.1 and Theorem 2.7 that there exist infinitely many $(q+1)$-tuples of the form $(N,{m}_{1},\mathrm{\dots},{m}_{q})\in {\mathbb{N}}^{q+1}$ such that, for any $1\le k\le q$, there holds

$i({c}_{k}^{2{m}_{k}-1})+\nu ({c}_{k}^{2{m}_{k}-1})=2N-(i({c}_{k})+2{S}_{{M}_{k}}^{+}(1)-\nu ({c}_{k})),$(3.2)$i({c}_{k}^{2{m}_{k}})\ge 2N-{\displaystyle \frac{e({P}_{{c}_{k}})}{2}},$(3.3)$i({c}_{k}^{2{m}_{k}})+\nu ({c}_{k}^{2{m}_{k}})\le 2N+{\displaystyle \frac{e({P}_{{c}_{k}})}{2}},$(3.4)$i({c}_{k}^{2{m}_{k}+1})=2N+i({c}_{k}).$(3.5)

Note that by [24, List 9.1.12] and the fact that $\nu ({c}_{k})={p}_{{k}_{-}}+2{p}_{{k}_{0}}+{p}_{{k}_{+}}$ we obtain

$2{S}_{{M}_{k}}^{+}(1)-\nu ({c}_{k})=2({p}_{{k}_{-}}+{p}_{{k}_{0}})-({p}_{{k}_{-}}+2{p}_{{k}_{0}}+{p}_{{k}_{+}})={p}_{{k}_{-}}-{p}_{{k}_{+}}.$(3.6)

So, by (3.1)–(3.6) and the fact that $e({P}_{{c}_{k}})\le 4$, we have

$i({c}_{k}^{m})+\nu ({c}_{k}^{m})\le 2N-i({c}_{k})-{p}_{{k}_{-}}+{p}_{{k}_{+}},$$\mathrm{\hspace{1em}}\text{for all}1\le m<2{m}_{k},$(3.7)$i({c}_{k}^{2{m}_{k}})+\nu ({c}_{k}^{2{m}_{k}})\le 2N+{\displaystyle \frac{e({P}_{{c}_{k}})}{2}}\le 2N+2,$(3.8)$2N+2\le i({c}_{k}^{m}),$$\mathrm{\hspace{1em}}\text{for all}m>2{m}_{k}.$(3.9)

In addition, the precise formulae of $i({c}_{k}^{2{m}_{k}})$ and $i({c}_{k}^{2{m}_{k}})+\nu ({c}_{k}^{2{m}_{k}})$ for $k=1,\mathrm{\dots},q$ can be computed as follows
(cf. [9, (3.16) and (3.21)] for the details):

$i({c}_{k}^{2{m}_{k}})=2N-{S}_{{M}_{k}}^{+}(1)-C({M}_{k})+2{\mathrm{\Delta}}_{k},$(3.10)$i({c}_{k}^{2{m}_{k}})+\nu ({c}_{k}^{2{m}_{k}})=2N+{p}_{{k}_{0}}+{p}_{{k}_{+}}+{q}_{{k}_{-}}+{q}_{{k}_{0}}$$+2{r}_{{k}_{0}}^{\prime}-2({r}_{{k}_{\ast}}-{r}_{{k}_{\ast}}^{\prime})+2{r}_{k}^{\prime}-{r}_{k}+2{\mathrm{\Delta}}_{k}$(3.11)

where ${r}_{k}$, ${r}_{{k}_{\ast}}$, and ${r}_{{k}_{0}}$ denote the number of normal forms $R(\theta )$, ${N}_{2}(\mathrm{exp}(\alpha \sqrt{-1}),A)$, and ${N}_{2}(\mathrm{exp}(\beta \sqrt{-1}),B)$ in (2.8) of Theorem 2.6 with $P={P}_{{c}_{k}}$, $k=1,2$, respectively, and ${r}_{k}^{\prime}$, ${r}_{{k}_{\ast}}^{\prime}$, and ${r}_{{k}_{0}}^{\prime}$ denote the number of normal forms $R(\theta )$, ${N}_{2}(\mathrm{exp}(\alpha \sqrt{-1}),A)$, and ${N}_{2}(\mathrm{exp}(\beta \sqrt{-1}),B)$ with $\theta ,\alpha ,\beta $ being the rational multiples of π in (2.8) of Theorem 2.6 with $P={P}_{{c}_{k}}$, $k=1,2$, respectively, and

${\mathrm{\Delta}}_{k}\equiv \sum _{0<\{{m}_{k}\theta /\pi \}<\delta}{S}_{{M}_{k}}^{-}(\mathrm{exp}(\theta \sqrt{-1}))\le {r}_{k}-{r}_{k}^{\prime}+{r}_{{k}_{\ast}}-{r}_{{k}_{\ast}}^{\prime},C({M}_{k})\equiv \sum _{\theta \in (0,2\pi )}{S}_{{M}_{k}}^{-}(\mathrm{exp}(\theta \sqrt{-1})),$(3.12)

where $\delta >0$ is a small enough number (cf. [27, (4.43)]) and the estimate of ${\mathrm{\Delta}}_{k}$ follows from the inequality [9, (3.18)].

Under the assumption (FCG), using [9, Theorem 1.1], we have that there exist at least two elliptic closed geodesics ${c}_{1}$ and ${c}_{2}$ on $({S}^{3},F)$ whose flag curvature satisfies ${(\lambda /(1+\lambda ))}^{2}<K\le 1$. The next lemma (cf. [9, Section 3]) lists some properties of these two closed geodesics which will be useful in the proof of Theorem 1.1.

*Under the assumption *(FCG)*, there exist at least two elliptic closed geodesics ${c}_{\mathrm{1}}$ and ${c}_{\mathrm{2}}$ on $\mathrm{(}{S}^{\mathrm{3}}\mathrm{,}F\mathrm{)}$ whose flag curvature satisfies ${\mathrm{(}\lambda \mathrm{/}\mathrm{(}\mathrm{1}\mathrm{+}\lambda \mathrm{)}\mathrm{)}}^{\mathrm{2}}\mathrm{<}K\mathrm{\le}\mathrm{1}$. Moreover, there exist infinitely many pairs of $\mathrm{(}q\mathrm{+}\mathrm{1}\mathrm{)}$-tuples of the form $\mathrm{(}N\mathrm{,}{m}_{\mathrm{1}}\mathrm{,}{m}_{\mathrm{2}}\mathrm{,}\mathrm{\dots}\mathrm{,}{m}_{q}\mathrm{)}\mathrm{\in}{\mathbb{N}}^{q\mathrm{+}\mathrm{1}}$ and $\mathrm{(}{N}^{\mathrm{\prime}}\mathrm{,}{m}_{\mathrm{1}}^{\mathrm{\prime}}\mathrm{,}{m}_{\mathrm{2}}^{\mathrm{\prime}}\mathrm{,}\mathrm{\dots}\mathrm{,}{m}_{q}^{\mathrm{\prime}}\mathrm{)}\mathrm{\in}{\mathbb{N}}^{q\mathrm{+}\mathrm{1}}$ such that*

$i({c}_{1}^{2{m}_{1}})+\nu ({c}_{1}^{2{m}_{1}})=2N+2,$$\mathrm{\hspace{0.25em}}{\overline{C}}_{2N+2}(E,{c}_{1}^{2{m}_{1}})=\mathbb{Q},$(3.13)$i({c}_{2}^{2{m}_{2}^{\prime}})+\nu ({c}_{2}^{2{m}_{2}^{\prime}})=2{N}^{\prime}+2,$${\overline{C}}_{2{N}^{\prime}+2}(E,{c}_{2}^{2{m}_{2}^{\prime}})=\mathbb{Q},$(3.14)

*and*

${p}_{{k}_{-}}={q}_{{k}_{+}}={r}_{{k}_{\ast}}={r}_{{k}_{0}}-{r}_{{k}_{0}}^{\prime}={h}_{k}=0,k=1,2,$(3.15)${r}_{1}-{r}_{1}^{\prime}={\mathrm{\Delta}}_{1}\ge 1,{r}_{2}-{r}_{2}^{\prime}={\mathrm{\Delta}}_{2}^{\prime}\ge 1,$(3.16)${\mathrm{\Delta}}_{k}+{\mathrm{\Delta}}_{k}^{\prime}={r}_{k}-{r}_{k}^{\prime},k=1,2,$(3.17)

*where we can require $\mathrm{2}\mathrm{|}N$ or $\mathrm{2}\mathrm{|}{N}^{\mathrm{\prime}}$ as remarked in Theorem 2.7 and
*

${\mathrm{\Delta}}_{k}^{\prime}\equiv \sum _{0<\{{m}_{k}^{\prime}\theta /\pi \}<\delta}{S}_{{M}_{k}}^{-}(\mathrm{exp}(\theta \sqrt{-1})),k=1,2.$(3.18)

#### Proof.

In fact, all these properties have already been obtained in [9, Section 3] and here we only list references. More precisely, (3.13) follows from [9, Claim 1] and the arguments between [9, (3.25) and (3.26)], (3.14) follows from [9, Claim 3] and similar arguments as those for ${c}_{1}$ between [9, (3.25) and (3.26)], (3.15) and (3.16) follow from [9, (3.25), Claim 2, and Claim 3], and, finally, (3.17) follows from [9, (3.31)] and (3.15). In one word, the properties of ${c}_{1}$ and ${c}_{2}$ are symmetric.
∎

*Under the assumption *(FCG)*, for the two elliptic closed geodesics ${c}_{\mathrm{1}}$, ${c}_{\mathrm{2}}$ found in Lemma 3.2, there holds*

${k}_{\nu ({c}_{k}^{n({c}_{k})})}^{\u03f5({c}_{k}^{n({c}_{k})})}({c}_{k}^{n({c}_{k})})=1,{k}_{j}^{\u03f5({c}_{k}^{n({c}_{k})})}({c}_{k}^{n({c}_{k})})=0$(3.19)

*for all $\mathrm{0}\mathrm{\le}j\mathrm{<}\nu \mathit{}\mathrm{(}{c}_{k}^{n\mathit{}\mathrm{(}{c}_{k}\mathrm{)}}\mathrm{)}$, $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}$, and then $\widehat{\chi}\mathit{}\mathrm{(}{c}_{k}\mathrm{)}\mathrm{\le}\mathrm{1}$ for $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}$.*

#### Proof.

We only give the proof for ${c}_{1}$. The proof for ${c}_{2}$ is identical.

First, by (3.13) and Lemma 2.1, we have

$1=dim{\overline{C}}_{2N+2}(E,{c}_{1}^{2{m}_{1}})$$=dim{H}_{2N+2-i({c}_{1}^{2{m}_{1}})}{({N}_{{c}_{1}^{2{m}_{1}}}\cup \{{c}_{1}^{2{m}_{1}}\},{N}_{{c}_{1}^{2{m}_{1}}})}^{\u03f5({c}_{1}^{2{m}_{1}}){\mathbb{Z}}_{2{m}_{1}}}$$=dim{H}_{\nu ({c}_{1}^{2{m}_{1}})}{({N}_{{c}_{1}^{2{m}_{1}}}\cup \{{c}_{1}^{2{m}_{1}}\},{N}_{{c}_{1}^{2{m}_{1}}})}^{\u03f5({c}_{1}^{2{m}_{1}}){\mathbb{Z}}_{2{m}_{1}}}$$={k}_{\nu ({c}_{1}^{2{m}_{1}})}^{\u03f5({c}_{1}^{2{m}_{1}})}({c}_{1}^{2{m}_{1}}),$

which implies that

${k}_{j}^{\u03f5({c}_{1}^{2{m}_{1}})}({c}_{1}^{2{m}_{1}})=0$

for any $0\le j<\nu ({c}_{1}^{2{m}_{1}})$ by Lemma 2.2 (ii). In addition, note that since $n({c}_{1})|2{m}_{1}$ and $\nu ({c}_{1}^{2{m}_{1}})=\nu ({c}_{1}^{n({c}_{1})})$ by (2.4) and (2.12), there holds

${k}_{j}^{\u03f5({c}_{1}^{2{m}_{1}})}({c}_{1}^{2{m}_{1}})={k}_{j}^{\u03f5({c}_{1}^{n({c}_{1})})}({c}_{1}^{n({c}_{1})})$

for any $0\le j\le \nu ({c}_{1}^{2{m}_{1}})$ by Lemma 2.2 (iv). Thus, (3.19) holds.

Note that by (3.16), the linearized Poincaré map ${P}_{{c}_{k}}$ of the elliptic closed geodesic ${c}_{k}$ is conjugate to $R({\theta}_{1})\diamond R({\theta}_{2})$ or $R({\theta}_{1})\diamond {N}_{1}(\lambda ,b)$ for some ${\theta}_{1}/2\pi \in (0,1)\backslash \mathbb{Q}$, $\lambda =\pm 1$, and $b=0,1$. Then,

$\nu ({c}_{k}^{m})=0$(3.20)

for all $m<n({c}_{k})$. In fact, when ${P}_{{c}_{k}}$ is conjugate to $R({\theta}_{1})\diamond {N}_{1}(1,b)$, we have $n({c}_{k})=1$. By (2.4), (3.20) holds. When ${P}_{{c}_{k}}$ is conjugate to $R({\theta}_{1})\diamond {N}_{1}(-1,b)$, we have $n({c}_{k})=2$. By (2.4), (3.20) also holds. When ${P}_{{c}_{k}}$ is conjugate to $R({\theta}_{1})\diamond R({\theta}_{2})$, (3.20) holds by (2.4).

Then, (3.20) yields

${k}_{0}^{\u03f5({c}_{k}^{m})}({c}_{k}^{m})=1,{k}_{j}^{\u03f5({c}_{k}^{m})}({c}_{k}^{m})=0$

for all $0<j\le 4$ and for $1\le m<n({c}_{k})$, which together with (2.3) and (3.19) gives

$\widehat{\chi}({c}_{k})=\frac{1}{n({c}_{k})}\left({(-1)}^{i({c}_{k}^{n({c}_{k})})+\nu ({c}_{k}^{n({c}_{k})})}+\sum _{1\le m<n({c}_{k})}{(-1)}^{i({c}_{k}^{m})}\right)\le 1.\mathit{\u220e}$

#### Proof of Theorem 1.1.

In order to prove Theorem 1.1, based on [33, Theorem 1.5] (cf. also [8, Theorem 1.1]), we make the assumption that

$\text{there exist exactly two elliptic distinct closed geodesics}{c}_{1}\text{,}{c}_{2}\text{possessing all properties listed}$$\text{in Lemmas 3.2 and 3.3 and a third closed geodesic}{c}_{3}\text{on}({S}^{3},F)\text{with reversibility}\lambda \text{and}$$\text{flag curvature}K\text{satisfying}(9/4){(\lambda /(1+\lambda ))}^{2}<K\le 1\text{with}\lambda <2\text{.}$

${c}_{1}^{m}$
* has no contribution to the Morse-type numbers ${M}_{\mathrm{2}\mathit{}N\mathrm{+}\mathrm{1}}$, ${M}_{\mathrm{2}\mathit{}N}$, and ${M}_{\mathrm{2}\mathit{}N\mathrm{-}\mathrm{1}}$ for any $m\mathrm{\in}\mathbb{N}$, ${c}_{\mathrm{2}}^{m}$ has possible contribution to the Morse-type numbers ${M}_{\mathrm{2}\mathit{}N\mathrm{+}\mathrm{1}}$, ${M}_{\mathrm{2}\mathit{}N}$, or ${M}_{\mathrm{2}\mathit{}N\mathrm{-}\mathrm{1}}$ only when $m\mathrm{=}\mathrm{2}\mathit{}{m}_{\mathrm{2}}$, and this time ${c}_{\mathrm{2}}^{\mathrm{2}\mathit{}{m}_{\mathrm{2}}}$ has no contribution to ${M}_{\mathrm{2}\mathit{}N\mathrm{+}\mathrm{1}}$ and ${M}_{\mathrm{2}\mathit{}N\mathrm{-}\mathrm{1}}$, but contributes at most one to ${M}_{\mathrm{2}\mathit{}N}$.*

In fact, by (3.16), for $k=1,2$, the linearized Poincaré map ${P}_{{c}_{k}}$ of the elliptic closed geodesic ${c}_{k}$ is conjugate to $R({\theta}_{1})\diamond R({\theta}_{2})$ or $R({\theta}_{1})\diamond {N}_{1}(\lambda ,b)$ for some ${\theta}_{1}/2\pi \in (0,1)\backslash \mathbb{Q}$, $\lambda =\pm 1$, and $b=0,1$. Combining this fact with Lemma 3.1 and (3.7), we have

$i({c}_{k}^{m})+\nu ({c}_{k}^{m})\le 2N-i({c}_{k})-{p}_{{k}_{-}}+{p}_{{k}_{+}}\le 2N-1$(3.21)

for $m<2{m}_{k}$, $k=1,2$, where the equality in (3.21) holds if and only if ${P}_{{c}_{k}}$ is conjugate to $R({\theta}_{1})\diamond {N}_{1}(1,-1)$ and $i({c}_{k})=2$, but $i({c}_{k})\in 2\mathbb{N}-1$ when ${P}_{{c}_{k}}$ is conjugate to $R({\theta}_{1})\diamond {N}_{1}(1,-1)$, thus the equality in (3.21) does not hold.
Then,

$i({c}_{k}^{m})+\nu ({c}_{k}^{m})\le 2N-2$(3.22)

for $m<2{m}_{k}$, $k=1,2$. Combining Lemma 2.2 (i) with (3.9) and (3.22), we know that ${c}_{k}^{m}$ has no contribution to the Morse-type numbers ${M}_{2N+1}$, ${M}_{2N}$, and ${M}_{2N-1}$ for $m\ne 2{m}_{k}$, where $k=1,2$. Note that by (3.13) and (3.19), ${c}_{1}^{2{m}_{1}}$ has also no contribution to ${M}_{2N+1}$, ${M}_{2N}$, and ${M}_{2N-1}$.

On one hand, there holds

$\nu ({c}_{2}^{2{m}_{2}})=\nu ({c}_{2}^{2{m}_{2}^{\prime}})$

by the choices of ${m}_{2}$ and ${m}_{2}^{\prime}$ in (2.12) of Theorem 2.7. On the other hand, it yields

$i({c}_{2}^{2{m}_{2}^{\prime}})=i({c}_{2}^{2{m}_{2}})(\mathrm{mod}\mathrm{\hspace{0.25em}2})$

by (2.10) of Theorem 2.6. So, $i({c}_{2}^{2{m}_{2}})+\nu ({c}_{2}^{2{m}_{2}})$ is even since $i({c}_{2}^{2{m}_{2}^{\prime}})+\nu ({c}_{2}^{2{m}_{2}^{\prime}})$ is even by (3.14) of Lemma 3.2, and then ${c}_{2}^{2{m}_{2}}$ has no contribution to ${M}_{2N+1}$ and ${M}_{2N-1}$ by (3.19). If ${c}_{2}^{2{m}_{2}}$ has contribution to ${M}_{2N}$, then ${c}_{2}^{2{m}_{2}}$ contributes exactly one to ${M}_{2N}$ by (3.19). Hence, Claim 1 holds.

${c}_{3}^{m}$
* has no contribution to the Morse-type numbers ${M}_{\mathrm{2}\mathit{}N\mathrm{+}\mathrm{1}}$, ${M}_{\mathrm{2}\mathit{}N}$, and ${M}_{\mathrm{2}\mathit{}N\mathrm{-}\mathrm{1}}$ for any $m\mathrm{\ne}\mathrm{2}\mathit{}{m}_{\mathrm{3}}$.*

First, by (3.9) and Lemma 2.2 (i), we know that ${c}_{3}^{m}$ has no contribution to the Morse-type numbers ${M}_{2N+1}$, ${M}_{2N}$, and ${M}_{2N-1}$ for $m>2{m}_{3}$.

On the other hand, from Lemma 3.1 and (3.1)–(3.2) along with the fact that $\nu ({c}_{3}^{2{m}_{3}-1})=\nu ({c}_{3})$, we have

$i({c}_{3}^{m})+\nu ({c}_{3}^{m})\le i({c}_{3}^{2{m}_{3}-1})=2N-(i({c}_{3})+2{S}_{{M}_{3}}^{+}(1))\le 2N-2$

for all $1\le m<2{m}_{3}-1$, which, together with Lemma 2.2 (i), implies that ${c}_{3}^{m}$ has no contribution to the Morse-type numbers ${M}_{2N+1}$, ${M}_{2N}$, and ${M}_{2N-1}$ for any $m<2{m}_{3}-1$.

Now, we prove Claim 2 by contradiction. We can assume that ${c}_{3}^{2{m}_{3}-1}$ has contribution to the Morse-type numbers ${M}_{2N+1}$, ${M}_{2N}$, or ${M}_{2N-1}$, i.e.,

$\sum _{q=2N-1}^{2N+1}dim{\overline{C}}_{q}(E,{c}^{2{m}_{3}-1})\ge 1.$(3.23)

Note that, by (3.2) and (3.6), we have

$i({c}_{3}^{2{m}_{3}-1})+\nu ({c}_{3}^{2{m}_{3}-1})=2N-i({c}_{3})-{p}_{{3}_{-}}+{p}_{{3}_{+}}\le 2N-2+2=2N,$(3.24)

which, together with Lemma 2.2 (i) and the assumption (3.23), gives $i({c}_{3}^{2{m}_{3}-1})+\nu ({c}_{3}^{2{m}_{3}-1})=2N$ or $2N-1$.

We continue the proof by distinguishing two cases.

*Case 1. $i\mathit{}\mathrm{(}{c}_{\mathrm{3}}^{\mathrm{2}\mathit{}{m}_{\mathrm{3}}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{+}\nu \mathit{}\mathrm{(}{c}_{\mathrm{3}}^{\mathrm{2}\mathit{}{m}_{\mathrm{3}}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{=}\mathrm{2}\mathit{}N$.* In this case, by (3.24), ${P}_{{c}_{3}}$ is conjugate to ${N}_{1}{(1,-1)}^{\diamond 2}$ and $i({c}_{3})=2$, $\nu ({c}_{3}^{m})=2$ for all $m\ge 1$. Then, using Theorem 2.6, we obtain

$i({c}_{3}^{m})+\nu ({c}_{3}^{m})-2=i({c}_{3}^{m})=mi({c}_{3})=m(i({c}_{3})+\nu ({c}_{3})-2)=2m$(3.25)

for all $m\ge 1$.

Now, by (2.5) and (2.7), we obtain ${M}_{2N}\ge {b}_{2N}=2$, which, together with Claim 1, implies that ${c}_{3}^{m}$ must have contribution to ${M}_{2N}$ for some $m\in \mathbb{N}$, i.e.,

$\sum _{m\ge 1}dim{\overline{C}}_{2N}(E,{c}_{3}^{m})\ge 1.$(3.26)

Thus, ${c}_{3}^{2{m}_{3}-1}$ has contribution to ${M}_{2N}$ and ${k}_{\nu ({c}_{3})}({c}_{3})=1$, since otherwise ${c}_{3}^{2{m}_{3}-1}$ contributes to ${M}_{2N-1}$ and ${k}_{1}({c}_{3})\ne 0$, and then ${c}_{3}^{m}$ has no contribution to ${M}_{2N}$ for any $m\in \mathbb{N}$ by (3.25), which contradicts (3.26). Now, ${k}_{\nu ({c}_{3})}({c}_{3})=1$ and (3.25) imply that ${c}_{3}$ satisfies the condition of Hingston’s result (cf. [17, Proposition 1] and [33, Theorem 4.2]), which yields the existence of infinitely many closed geodesics which contradicts the assumption (TCG).

*Case 2. $i\mathit{}\mathrm{(}{c}_{\mathrm{3}}^{\mathrm{2}\mathit{}{m}_{\mathrm{3}}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{+}\nu \mathit{}\mathrm{(}{c}_{\mathrm{3}}^{\mathrm{2}\mathit{}{m}_{\mathrm{3}}\mathrm{-}\mathrm{1}}\mathrm{)}\mathrm{=}\mathrm{2}\mathit{}N\mathrm{-}\mathrm{1}$.* In this case, by (3.24), one of the following cases may happen.

For (i), we have that ${P}_{{c}_{3}}$ is conjugate to ${N}_{1}{(1,-1)}^{\diamond 2}$, which implies that $i({c}_{3})$ is even, thus case (i) cannot happen.

Noticing that $i({c}_{3})=2$ is even in case (ii), we have that ${P}_{{c}_{3}}$ is conjugate to ${N}_{1}(1,-1)\diamond H(2)$. So, by Theorem 2.6, we have

$i({c}_{3}^{m})+\nu ({c}_{3}^{m})=mi({c}_{3})+\nu ({c}_{3}^{m})=2m+1$(3.27)

for $m\ge 1$. Now, in this case it follows from (3.23) that ${c}_{3}^{2{m}_{3}-1}$ has contribution to ${M}_{2N-1}$ and then ${k}_{\nu ({c}_{3})}({c}_{3})=1$, which together with (3.27) implies that ${c}_{3}^{m}$ has no contribution to ${M}_{2N}$ for any $m\in \mathbb{N}$, which in turn contradicts (3.26). This completes the proof of Claim 2.

${c}_{2}^{2{m}_{2}}$
* has no contribution to ${M}_{\mathrm{2}\mathit{}N}$.*

In fact, ${c}_{2}^{2{m}_{2}}$ contributes otherwise exactly one to ${M}_{2N}$ by Claim 1. By (2.5) and (2.7), ${M}_{2N}\ge {b}_{2N}=2$, and then ${c}_{3}^{2{m}_{3}}$
must have contribution to ${M}_{2N}$ by Claims 1 and 2. Thus, ${c}_{3}^{2{m}_{3}}$ has no contribution to ${M}_{2N+2}$ and ${M}_{2N-2}$ by (3.3)–(3.4) and Lemma 2.2 (ii). So, we obtain that

$-{M}_{2N+1}+{M}_{2N}-{M}_{2N-1}=\sum _{0\le l\le 4}{(-1)}^{i({c}_{3}^{2{m}_{3}})+l}{k}_{l}^{\u03f5({c}_{3}^{2{m}_{3}})}({c}_{3}^{2{m}_{3}})+1.$(3.28)

On the other hand, by (2.6) and Lemma 2.4, we have

${M}_{2N+1}-{M}_{2N}+{M}_{2N-1}\ge {b}_{2N+1}-{b}_{2N}+{b}_{2N-1}=-2.$(3.29)

Combining (3.28) and (3.29), we get

$\chi ({c}_{3}^{2{m}_{3}})=\sum _{0\le l\le 4}{(-1)}^{i({c}_{3}^{2{m}_{3}})+l}{k}_{l}^{\u03f5({c}_{3}^{2{m}_{3}})}({c}_{3}^{2{m}_{3}})\le 1.$(3.30)

Note that since $n({c}_{3})|2{m}_{3}$ and $\nu ({c}_{3}^{2{m}_{3}})=\nu ({c}_{3}^{n({c}_{3})})$ by (2.4) and (2.12),
there holds

${k}_{j}^{\u03f5({c}_{3}^{2{m}_{3}})}({c}_{3}^{2{m}_{3}})={k}_{j}^{\u03f5({c}_{3}^{n({c}_{3})})}({c}_{3}^{n({c}_{3})})$

for any $0\le j\le \nu ({c}_{3}^{2{m}_{3}})$ by Lemma 2.2 (iv). Then, it follows from (2.4) and (3.30) that

$\chi ({c}_{3}^{n({c}_{3})})=\chi ({c}_{3}^{2{m}_{3}})\le 1.$(3.31)

Now, we can obtain that

$\chi ({c}_{3}^{m})\le 1$(3.32)

for all $1\le m<n({c}_{3})$.

In fact, if ${c}_{3}$ is totally degenerate, i.e., if 1 is the unique eigenvalue of ${P}_{{c}_{3}}$, then $n({c}_{3})=1$ and (3.32) holds by (3.31).

If ${c}_{3}$ is not totally degenerate, by (2.4), either $\nu ({c}_{3}^{m})<2$ for $1\le m<n({c}_{3})$ or $\nu ({c}_{3}^{{m}_{0}})=2$ for $1\le {m}_{0}<n({c}_{3})$ with ${P}_{{c}_{3}^{{m}_{0}}}$ conjugating to $I\diamond R(\theta )$ for some $\theta /2\pi \in \mathbb{Q}$ and $i({c}_{3}^{{m}_{0}})\in 2\mathbb{N}$. In any case, (3.32) follows from Lemma 2.2 (ii).

Now, we combine (3.31) and (3.32) to get $\widehat{\chi}({c}_{3})\le 1$, which, together with Lemma 3.1 and Lemma 3.3, implies that

$\sum _{j=1}^{3}\frac{\widehat{\chi}({c}_{j})}{\widehat{\u0131}({c}_{j})}<\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1,$

which contradicts the identity (2.2) in Theorem 2.3. Hence, Claim 3 holds.

${c}_{1}$
* and ${c}_{\mathrm{2}}$ are irrationally elliptic.*

By (3.16) and (3.17), there holds ${\mathrm{\Delta}}_{2}=0$. Then, together with the fact that ${r}_{{2}_{\ast}}=0$ from (3.15), it follows from (3.16) and (3.11) that

$2N\ge i({c}_{2}^{2{m}_{2}})+\nu ({c}_{2}^{2{m}_{2}})$(3.33)$=2N+({p}_{{2}_{0}}+{p}_{2+}+{q}_{{2}_{-}}+{q}_{{2}_{0}}+2{r}_{{2}_{0}}^{\prime}+{r}_{2}^{\prime})-({r}_{2}-{r}_{2}^{\prime})$$\ge 2N-2,$(3.34)

where (3.33) holds by the fact that ${p}_{{2}_{0}}+{p}_{2+}+{q}_{{2}_{-}}+{q}_{{2}_{0}}+2{r}_{{2}_{0}}^{\prime}+{r}_{2}^{\prime}\le 1$ from (2.9) and (3.16) and ${r}_{2}-{r}_{2}^{\prime}\ge 1$ from (3.16), and the equality in (3.34) holds if and only if ${r}_{2}-{r}_{2}^{\prime}=2$. On the other hand, by Claim 3, we have $i({c}_{2}^{2{m}_{2}})+\nu ({c}_{2}^{2{m}_{2}})\ne 2N$ and by (3.14), we have that

$i({c}_{2}^{2{m}_{2}})+\nu ({c}_{2}^{2{m}_{2}})$

is even since it has the same parity with $i({c}_{2}^{2{m}_{2}^{\prime}})+\nu ({c}_{2}^{2{m}_{2}^{\prime}})$. Thus, by (3.33), we obtain $i({c}_{2}^{2{m}_{2}})+\nu ({c}_{2}^{2{m}_{2}})\le 2N-2$, which together with (3.34) implies ${r}_{2}-{r}_{2}^{\prime}=2$, i.e., ${c}_{2}$ is irrationally elliptic. By the symmetry of ${c}_{1}$ and ${c}_{2}$, we also obtain that ${c}_{1}$ is irrationally elliptic. Thus, Claim 4 is true.

To conclude with the proof of Theorem 1.1, first note that if 1 is an eigenvalue of ${P}_{{c}_{3}^{{m}_{0}}}$ for some ${m}_{0}\in \mathbb{N}$, then 1 must be an eigenvalue of ${P}_{{c}_{3}^{2l{m}_{0}}}$ for any $l\in \mathbb{N}$ by (2.11) of Theorem 2.6. So, if ${c}_{3}$ is not infinitely degenerate, then all iterates ${c}_{3}^{m}$ of ${c}_{3}$ with $m\in \mathbb{N}$ are non-degenerate and then all closed geodesics ${c}_{k}$, $k=1,2,3$, and their iterates are non-degenerate by Claim 4. Using [35, Theorem 1.2], we get four prime closed geodesics, which contradicts the assumption (TCG). Hence, ${c}_{3}$ is infinitely degenerate.
∎

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