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Volume 16, Issue 1

Closed Geodesics on Positively Curved Finsler 3-Spheres

Huagui Duan
/ Hui Liu
• Corresponding author
• School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
• Email
• Other articles by this author:
Published Online: 2016-01-14 | DOI: https://doi.org/10.1515/ans-2015-5007

Abstract

In [33], Wang proved that for every Finsler three-dimensional sphere $\left({S}^{3},F\right)$ with reversibility λ and flag curvature K satisfying ${\left(\lambda /\left(1+\lambda \right)\right)}^{2}, there exist at least three distinct closed geodesics. In this paper, we prove that for every Finsler three-dimensional sphere $\left({S}^{3},F\right)$ with reversibility λ and flag curvature K satisfying $\left(9/4\right){\left(\lambda /\left(1+\lambda \right)\right)}^{2} with $\lambda <2$, if there exist exactly three prime closed geodesics, then two of them are irrationally elliptic and the third one is infinitely degenerate.

MSC 2010: 53C22; 58E05; 58E10

1 Introduction and Main Results

A closed curve on a Finsler manifold is a closed geodesic if it is locally the shortest path connecting any two nearby points on this curve. As usual, on any Finsler manifold $\left(M,F\right)$, a closed geodesic $c:{S}^{1}=ℝ/ℤ\to M$ 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 ${c}^{m}\left(t\right)=c\left(mt\right)$. The inverse curve ${c}^{-1}$ of c is defined by ${c}^{-1}\left(t\right)=c\left(1-t\right)$ for $t\in ℝ$. Note that unlike on Riemannian manifolds, the inverse curve ${c}^{-1}$ of a closed geodesic c on an irreversible Finsler manifold need not be a geodesic. We call two prime closed geodesics c and d distinct if there is no $\theta \in \left(0,1\right)$ such that $c\left(t\right)=d\left(t+\theta \right)$ for all $t\in ℝ$. On a reversible Finsler (or Riemannian) manifold, two closed geodesics c and d are called geometrically distinct if $c\left({S}^{1}\right)\ne d\left({S}^{1}\right)$, i.e., if their image sets in M are distinct. We shall omit the word distinct when we talk about more than one prime closed geodesic.

For a closed geodesic c on an n-dimensional manifold $\left(M,F\right)$, denote by ${P}_{c}$ the linearized Poincaré map of c. Then, ${P}_{c}\in Sp\left(2n-2\right)$ is symplectic. For any $M\in Sp\left(2k\right)$, we define the elliptic height $e\left(M\right)$ of M to be the total algebraic multiplicity of all eigenvalues of M on the unit circle $𝕌=\left\{z\in ℂ\mid |z|=1\right\}$ in the complex plane $ℂ$. Since M is symplectic, $e\left(M\right)$ is even and $0\le e\left(M\right)\le 2k$. A closed geodesic c is called elliptic if $e\left({P}_{c}\right)=2\left(n-1\right)$, i.e., if all the eigenvalues of ${P}_{c}$ are located on $𝕌$, irrationally elliptic if it is elliptic and ${P}_{c}$ is suitably homotopic to the $\diamond$-product of $n-1$ rotation $\left(2×2\right)$ matrices with rotation angles being irrational multiples of π, hyperbolic if $e\left({P}_{c}\right)=0$, i.e., all the eigenvalues of ${P}_{c}$ are located away from $𝕌$, infinitely degenerate if 1 is an eigenvalue of ${P}_{{c}^{m}}$ for infinitely many $m\in ℕ$, and, finally, non-degenerate if 1 is not an eigenvalue of ${P}_{c}$. A Finsler manifold $\left(M,F\right)$ is called bumpy if all the closed geodesics on it are non-degenerate.

There is a famous conjecture in Riemannian geometry which claims that there exist infinitely many closed geodesics on any compact Riemannian manifold. This conjecture has been proved except for compact rank-one symmetric spaces. The results of Franks [15] and Bangert [4] imply that this conjecture is true for any Riemannian 2-sphere (cf. [17, 18]). But once one moves to the Finsler case, the conjecture becomes false. It was quite surprising when Katok [19] found some irreversible Finsler metrics on spheres with only finitely many closed geodesics and all closed geodesics being non-degenerate and elliptic (cf. [37]).

Recently, index iteration theory of closed geodesics (cf. [6, 22, 23, 24]) has been applied to study the closed geodesic problem on Finsler manifolds. For example, Bangert and Long show in [5] that there exist at least two closed geodesics on every $\left({S}^{2},F\right)$. After that, a great number of multiplicity and stability results has appeared (cf. [10, 11, 12, 13, 14, 25, 26, 31, 32, 33, 35, 34, 36] and the references therein).

In [30], Rademacher has introduced the reversibility $\lambda =\lambda \left(M,F\right)$ of a compact Finsler manifold as

$\lambda =\mathrm{max}\left\{F\left(-X\right)|X\in TM,F\left(X\right)=1\right\}\ge 1.$

Then, in [31], he obtained some results on the multiplicity and the length of closed geodesics and their stability properties. For example, letting F be a Finsler metric on ${S}^{n}$ with reversibility λ and flag curvature K satisfying ${\left(\lambda /\left(1+\lambda \right)\right)}^{2}, there exist at least $n/2-1$ closed geodesics with length $<2n\pi$. If $\left(9/4\right){\left(\lambda /\left(1+\lambda \right)\right)}^{2} with $\lambda <2$, then there exists an elliptic-parabolic closed geodesic, i.e., its linearized Poincaré map is split into two-dimensional rotations and a part whose eigenvalues are $±1$. Some similar results in the Riemannian case are obtained in [2, 3].

Recently, Wang proved in [33] that for every Finsler n-dimensional sphere ${S}^{n}$ with reversibility λ and flag curvature K satisfying ${\left(\lambda /\left(1+\lambda \right)\right)}^{2}, either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form $\mathrm{exp}\left(\sqrt{-1}\pi \mu \right)$ with an irrational μ. The same author proved in [36] that for every Finsler n-dimensional sphere ${S}^{n}$ for $n\ge 6$ with reversibility λ and flag curvature K satisfying ${\left(\lambda /\left(1+\lambda \right)\right)}^{2}, either there exist infinitely many prime closed geodesics or there exist $\left[n/2\right]-2$ closed geodesics possessing irrational mean indices. Furthermore, assuming that the metric F is bumpy, he showed in [35] that there exist $2\left[\left(n+1\right)/2\right]$ closed geodesics on $\left({S}^{n},F\right)$. Also, in [35], he showed that for every bumpy Finsler metric F on ${S}^{n}$ satisfying $\left(9/4\right){\left(\lambda /\left(1+\lambda \right)\right)}^{2}, there exist two prime elliptic closed geodesics provided the number of closed geodesics on $\left({S}^{n},F\right)$ is finite.

Very recently, Duan proved in [9] that for every Finsler n-dimensional sphere $\left({S}^{n},F\right)$, $n\ge 2$, with reversibility λ and flag curvature K satisfying ${\left(\lambda /\left(1+\lambda \right)\right)}^{2}, either there exist infinitely many closed geodesics or there exist at least two elliptic closed geodesics and each linearized Poincaré map has at least one eigenvalue of the form $\mathrm{exp}\left(\theta \sqrt{-1}\right)$ with θ being an irrational multiple of π. Furthermore, in [8], he proved that for every Finsler metric F on the n-dimensional sphere ${S}^{n}$, $n\ge 3$, with reversibility λ and flag curvature K satisfying ${\left(\lambda /\left(1+\lambda \right)\right)}^{2}, either there exist infinitely many closed geodesics or there exist always three prime closed geodesics and at least two of them are elliptic; when $n\ge 6$, these three distinct closed geodesics are non-hyperbolic. If the metric is bumpy, Duan and Long proved in [11] that on every bumpy Finsler three-dimensional sphere $\left({S}^{3},F\right)$, either there exist two non-hyperbolic prime closed geodesics or there exist at least three prime closed geodesics.

Note that Wang proved in [33, Theorem 1.5] that there exist at least three distinct closed geodesics on $\left({S}^{3},F\right)$ with flag curvature K satisfying ${\left(\lambda /\left(1+\lambda \right)\right)}^{2}. Motivated by the results mentioned above, in this paper, we prove the following theorem.

For every Finsler metric F on the three-dimensional sphere ${S}^{\mathrm{3}}$ with reversibility λ and flag curvature K satisfying $\mathrm{\left(}\mathrm{9}\mathrm{/}\mathrm{4}\mathrm{\right)}\mathit{}{\mathrm{\left(}\lambda \mathrm{/}\mathrm{\left(}\mathrm{1}\mathrm{+}\lambda \mathrm{\right)}\mathrm{\right)}}^{\mathrm{2}}\mathrm{<}K\mathrm{\le }\mathrm{1}$ with $\lambda \mathrm{<}\mathrm{2}$, if there exist exactly three prime closed geodesics, then two of them are irrationally elliptic and the third one is infinitely degenerate.

Note that Anosov conjectured in [1] that the lower bound of the number of distinct closed geodesics on a Finsler three-dimensional sphere $\left({S}^{3},F\right)$ is four, where Katok’s examples in [19] show that this lower bound can be attained. However, Ziller in [37, pp. 155–156] conjectured that the lower bound of the number of distinct closed geodesics on a Finsler three-dimensional sphere $\left({S}^{3},F\right)$ is three. To our knowledge, it is not clear whether there exist some irreversible Finsler metrics on ${S}^{3}$ with exactly three distinct closed geodesics. This is an interesting problem.

Our proof of Theorem 1.1 in Section 3 contains mainly three ingredients: the common index jump theorem of [27], Morse theory, and some new symmetric information about the index jump. In addition, we also follow some ideas from our recent preprints [9, 8, 13].

In this paper, let $ℕ$, ${ℕ}_{0}$, $ℤ$, $ℚ$, $ℝ$, and $ℂ$ 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 $ℚ$-coefficients. For an ${S}^{1}$-space X, we denote by $\overline{X}$ the quotient space $X/{S}^{1}$. We define the functions

$\left[a\right]=\mathrm{max}\left\{k\in ℤ|k\le a\right\},E\left(a\right)=\mathrm{min}\left\{k\in ℤ|k\ge a\right\},\phi \left(a\right)=E\left(a\right)-\left[a\right],\left\{a\right\}=a-\left[a\right].$(1.1)

In particular, we have $\phi \left(a\right)=0$ if $a\in ℤ$ and $\phi \left(a\right)=1$ if $a\notin ℤ$.

2.1 Morse Theory for Closed Geodesics

Let $M=\left(M,F\right)$ be a compact Finsler manifold. Then, the space $\mathrm{\Lambda }=\mathrm{\Lambda }M$ of ${H}^{1}$-maps $\gamma :{S}^{1}\to M$ has a natural structure of Riemannian Hilbert manifolds on which the group ${S}^{1}=ℝ/ℤ$ acts continuously by isometries (cf. [20]). This action is defined by $\left(s\cdot \gamma \right)\left(t\right)=\gamma \left(t+s\right)$ for all $\gamma \in \mathrm{\Lambda }$ and $s,t\in {S}^{1}$. For any $\gamma \in \mathrm{\Lambda }$, the energy functional is defined by

$E\left(\gamma \right)=\frac{1}{2}{\int }_{{S}^{1}}F{\left(\gamma \left(t\right),\stackrel{˙}{\gamma }\left(t\right)\right)}^{2}𝑑t$(2.1)

and is ${C}^{1,1}$ and invariant under the ${S}^{1}$-action. The critical points of E of positive energies are precisely the closed geodesics $\gamma :{S}^{1}\to M$. The index form of the functional E is well-defined along any closed geodesic c on M, which we denote by ${E}^{\mathrm{\prime \prime }}\left(c\right)$. As usual, we denote by $i\left(c\right)$ and $\nu \left(c\right)-1$ the Morse index and the nullity of E at c, respectively. In the following, we denote

${\mathrm{\Lambda }}^{\kappa }=\left\{d\in \mathrm{\Lambda }|E\left(d\right)\le \kappa \right\},{\mathrm{\Lambda }}^{\kappa -}=\left\{d\in \mathrm{\Lambda }|E\left(d\right)<\kappa \right\}$

for all $\kappa \ge 0$. For a closed geodesic c, we set $\mathrm{\Lambda }\left(c\right)=\left\{\gamma \in \mathrm{\Lambda }\mid E\left(\gamma \right).

Recall that the mean index $\stackrel{^}{ı}\left(c\right)$ and the ${S}^{1}$-critical modules of ${c}^{m}$ are defined by

$\stackrel{^}{ı}\left(c\right)=\underset{m\to \mathrm{\infty }}{lim}\frac{i\left({c}^{m}\right)}{m},{\overline{C}}_{*}\left(E,{c}^{m}\right)={H}_{*}\left(\left(\mathrm{\Lambda }\left({c}^{m}\right)\cup {S}^{1}\cdot {c}^{m}\right)/{S}^{1},\mathrm{\Lambda }\left({c}^{m}\right)/{S}^{1};ℚ\right),$

respectively.

We say that a closed geodesic satisfies the isolation condition if

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

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

If c has multiplicity m, then the subgroup ${ℤ}_{m}=\left\{n/m\mid 0\le n of ${S}^{1}$ acts on ${\overline{C}}_{*}\left(E,c\right)$. As studied in [29, p. 59], for all $m\in ℕ$, let ${H}_{\ast }{\left(X,A\right)}^{±{ℤ}_{m}}=\left\{\left[\xi \right]\in {H}_{\ast }\left(X,A\right)\mid {T}_{\ast }\left[\xi \right]=±\left[\xi \right]\right\}$, where T is a generator of the ${ℤ}_{m}$-action. On ${S}^{1}$-critical modules of ${c}^{m}$, the following lemma holds (cf. [29, Satz 6.11], [5]).

Suppose c is a prime closed geodesic on a Finsler manifold M satisfying (Iso). Then, there exist ${U}_{{c}^{m}}$ and ${N}_{{c}^{m}}$, the so-called local negative disk and the local characteristic manifold at ${c}^{m}$, respectively, such that $\nu \mathit{}\mathrm{\left(}{c}^{m}\mathrm{\right)}\mathrm{=}\mathrm{dim}\mathit{}{N}_{{c}^{m}}$ and

${\overline{C}}_{q}\left(E,{c}^{m}\right)\equiv {H}_{q}\left(\left(\mathrm{\Lambda }\left({c}^{m}\right)\cup {S}^{1}\cdot {c}^{m}\right)/{S}^{1},\mathrm{\Lambda }\left({c}^{m}\right)/{S}^{1}\right)$$={\left({H}_{i\left({c}^{m}\right)}\left({U}_{{c}^{m}}^{-}\cup \left\{{c}^{m}\right\},{U}_{{c}^{m}}^{-}\right)\otimes {H}_{q-i\left({c}^{m}\right)}\left({N}_{{c}^{m}}^{-}\cup \left\{{c}^{m}\right\},{N}_{{c}^{m}}^{-}\right)\right)}^{+{ℤ}_{m}},$

where ${U}_{{c}^{m}}^{\mathrm{-}}\mathrm{=}{U}_{{c}^{m}}\mathrm{\cap }\mathrm{\Lambda }\mathit{}\mathrm{\left(}{c}^{m}\mathrm{\right)}$, ${N}_{{c}^{m}}^{\mathrm{-}}\mathrm{=}{N}_{{c}^{m}}\mathrm{\cap }\mathrm{\Lambda }\mathit{}\mathrm{\left(}{c}^{m}\mathrm{\right)}$.

• (i)

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

${\overline{C}}_{q}\left(E,{c}^{m}\right)=\left\{\begin{array}{cc}ℚ\hfill & \mathit{\text{if}}i\left({c}^{m}\right)-i\left(c\right)\in 2ℤ\mathit{\text{and}}q=i\left({c}^{m}\right),\hfill \\ 0\hfill & \mathit{\text{otherwise.}}\hfill \end{array}$

• (ii)

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

${\overline{C}}_{q}\left(E,{c}^{m}\right)={H}_{q-i\left({c}^{m}\right)}{\left({N}_{{c}^{m}}^{-}\cup \left\{{c}^{m}\right\},{N}_{{c}^{m}}^{-}\right)}^{ϵ\left({c}^{m}\right){ℤ}_{m}},$

where $ϵ\left({c}^{m}\right)={\left(-1\right)}^{i\left({c}^{m}\right)-i\left(c\right)}$.

Define

${k}_{j}\left({c}^{m}\right)\equiv dim{H}_{j}\left({N}_{{c}^{m}}^{-}\cup \left\{{c}^{m}\right\},{N}_{{c}^{m}}^{-}\right),{k}_{j}^{±1}\left({c}^{m}\right)\equiv dim{H}_{j}{\left({N}_{{c}^{m}}^{-}\cup \left\{{c}^{m}\right\},{N}_{{c}^{m}}^{-}\right)}^{±{ℤ}_{m}}.$

Then, we have the following lemma (cf. [29, 26, 33]).

Let c be a prime closed geodesic on a Finsler manifold $\mathrm{\left(}M\mathrm{,}F\mathrm{\right)}$. Then, we have the following.

• (i)

For any $m\in ℕ$ , there holds ${k}_{j}\left({c}^{m}\right)=0$ for $j\ne \left[0,\nu \left({c}^{m}\right)\right]$.

• (ii)

For any $m\in ℕ$ , there holds ${k}_{0}\left({c}^{m}\right)+{k}_{\nu \left({c}^{m}\right)}\left({c}^{m}\right)\le 1$ and if ${k}_{0}\left({c}^{m}\right)+{k}_{\nu \left({c}^{m}\right)}\left({c}^{m}\right)=1$ , then there holds ${k}_{j}\left({c}^{m}\right)=0$ for $j\in \left(0,\nu \left({c}^{m}\right)\right)$.

• (iii)

For any $m\in ℕ$ , there holds ${k}_{0}^{+1}\left({c}^{m}\right)={k}_{0}\left({c}^{m}\right)$ and ${k}_{0}^{-1}\left({c}^{m}\right)=0$ . In particular, if ${c}^{m}$ is non-degenerate, then there holds ${k}_{0}^{+1}\left({c}^{m}\right)={k}_{0}\left({c}^{m}\right)=1$ and ${k}_{0}^{-1}\left({c}^{m}\right)={k}_{j}^{±1}\left({c}^{m}\right)=0$ for all $j\ne 0$.

• (iv)

Suppose that the nullities satisfy $\nu \left({c}^{m}\right)=\nu \left({c}^{n}\right)$ for some integer $m=np\ge 2$ with $n,p\in ℕ$ . Then, there holds ${k}_{j}\left({c}^{m}\right)={k}_{j}\left({c}^{n}\right)$ and ${k}_{j}^{±1}\left({c}^{m}\right)={k}_{j}^{±1}\left({c}^{n}\right)$ for any integer j.

Let $\left(M,F\right)$ be a compact simply connected Finsler manifold with finitely many closed geodesics. Denote those prime closed geodesics on $\left(M,F\right)$ with positive mean indices by ${\left\{{c}_{j}\right\}}_{1\le j\le k}$. Rademacher established in [28, 29] a celebrated mean index identity relating all ${c}_{j}$ with the global homology of M (cf. [29, Section 7], especially Satz 7.9 therein) for compact simply connected Finsler manifolds. Here, we give a brief review on this identity (cf. [29, Satz 7.9] and also [12, 26, 33]).

Assume that there exist finitely many closed geodesics on $\mathrm{\left(}{S}^{\mathrm{3}}\mathrm{,}F\mathrm{\right)}$ and denote the prime closed geodesics with positive mean indices by ${\mathrm{\left\{}{c}_{j}\mathrm{\right\}}}_{\mathrm{1}\mathrm{\le }j\mathrm{\le }k}$ for some $k\mathrm{\in }ℕ$. Then, we have the identity

$\sum _{j=1}^{k}\frac{\stackrel{^}{\chi }\left({c}_{j}\right)}{\stackrel{^}{ı}\left({c}_{j}\right)}=1,$(2.2)

where

$\stackrel{^}{\chi }\left({c}_{j}\right)=\frac{1}{n\left({c}_{j}\right)}\sum _{\begin{array}{c}1\le m\le n\left({c}_{j}\right)\\ 0\le l\le 2\left(n-1\right)\end{array}}\chi \left({c}_{j}^{m}\right)=\frac{1}{n\left({c}_{j}\right)}\sum _{\begin{array}{c}1\le m\le n\left({c}_{j}\right)\\ 0\le l\le 4\end{array}}{\left(-1\right)}^{i\left({c}_{j}^{m}\right)+l}{k}_{l}^{ϵ\left({c}_{j}^{m}\right)}\left({c}_{j}^{m}\right)\in ℚ$(2.3)

and the analytical period $n\mathit{}\mathrm{\left(}{c}_{j}\mathrm{\right)}$ of ${c}_{j}$ is defined by (cf. [26] )

$n\left({c}_{j}\right)=\mathrm{min}\left\{l\in ℕ|\nu \left({c}_{j}^{l}\right)=\underset{m\ge 1}{\mathrm{max}}\nu \left({c}_{j}^{m}\right)\mathit{\text{with}}i\left({c}_{j}^{m+l}\right)-i\left({c}_{j}^{m}\right)\in 2ℤ\mathit{\text{for all}}m\in ℕ\right\}.$(2.4)

Set

${\overline{\mathrm{\Lambda }}}^{0}={\overline{\mathrm{\Lambda }}}^{0}{S}^{3}=\left\{\text{constant point curves in}{S}^{3}\right\}\cong {S}^{3}.$

Let $\left(X,Y\right)$ be a space pair such that the Betti numbers ${b}_{i}={b}_{i}\left(X,Y\right)=dim{H}_{i}\left(X,Y;ℚ\right)$ are finite for all $i\in ℤ$. As usual, the Poincaré series of $\left(X,Y\right)$ is defined by the formal power series $P\left(X,Y\right)={\sum }_{i=0}^{\mathrm{\infty }}{b}_{i}{t}^{i}$. We need the following well-known version of results on Betti numbers and the Morse inequality. For Lemma 2.4 below, see [28, Theorem 2.4 and Remark 2.5], [16], and also [12, Lemma 2.5]), and for Theorem 2.5, see [7, Theorem I.4.3].

Let $\mathrm{\left(}{S}^{\mathrm{3}}\mathrm{,}F\mathrm{\right)}$ be a three-dimensional Finsler sphere. Then, the Betti numbers are given by

${b}_{j}=rank{H}_{j}\left(\mathrm{\Lambda }{S}^{3}/{S}^{1},{\mathrm{\Lambda }}^{0}{S}^{3}/{S}^{1};ℚ\right)=\left\{\begin{array}{cc}2\hfill & \mathit{\text{if}}j=2k\ge 4,\hfill \\ 1\hfill & \mathit{\text{if}}j=2,\hfill \\ 0\hfill & \text{𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒}.\hfill \end{array}$(2.5)

Let $\mathrm{\left(}M\mathrm{,}F\mathrm{\right)}$ be a Finsler manifold with finitely many closed geodesics, denoted by ${\mathrm{\left\{}{c}_{j}\mathrm{\right\}}}_{\mathrm{1}\mathrm{\le }j\mathrm{\le }k}$. Set

${M}_{q}=\sum _{\begin{array}{c}1\le j\le k\\ m\ge 1\end{array}}dim{\overline{C}}_{q}\left(E,{c}_{j}^{m}\right),q\in ℤ.$

Then, for every integer $q\mathrm{\ge }\mathrm{0}$, there holds

${M}_{q}-{M}_{q-1}+\mathrm{\cdots }+{\left(-1\right)}^{q}{M}_{0}\ge {b}_{q}-{b}_{q-1}+\mathrm{\cdots }+{\left(-1\right)}^{q}{b}_{0}$(2.6)${M}_{q}\ge {b}_{q}.$(2.7)

2.2 Index Iteration Theory of Closed Geodesics

In [22], Long established the basic normal form decomposition of symplectic matrices. Based on this result, he further established the precise iteration formulae of indices of symplectic paths in [23]. Note that this index iteration formulae works for Morse indices of iterated closed geodesics (cf. [21] and [24, Chapter 12]). Since every closed geodesic on a sphere must be orientable, then, by [21, Theorem 1.1], the initial Morse index of a closed geodesic on a Finsler ${S}^{n}$ coincides with the index of a corresponding symplectic path.

As in [23], we denote

${N}_{1}\left(\lambda ,b\right)=\left(\begin{array}{cc}\hfill \lambda \hfill & \hfill b\hfill \\ \hfill 0\hfill & \hfill \lambda \hfill \end{array}\right),\lambda =±1,b\in ℝ,$$H\left(\lambda \right)=\left(\begin{array}{cc}\hfill \lambda \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\lambda }^{-1}\hfill \end{array}\right),\lambda \in ℝ\setminus \left\{0,±1\right\},$$R\left(\theta \right)=\left(\begin{array}{cc}\hfill \mathrm{cos}\theta \hfill & \hfill -\mathrm{sin}\theta \hfill \\ \hfill \mathrm{sin}\theta \hfill & \hfill \mathrm{cos}\theta \hfill \end{array}\right),\theta \in \left(0,\pi \right)\cup \left(\pi ,2\pi \right),$

and

${N}_{2}\left(\mathrm{exp}\left(\theta \sqrt{-1}\right),B\right)=\left(\begin{array}{cc}\hfill R\left(\theta \right)\hfill & \hfill B\hfill \\ \hfill 0\hfill & \hfill R\left(\theta \right)\hfill \end{array}\right),\theta \in \left(0,\pi \right)\cup \left(\pi ,2\pi \right),$

where

$B=\left(\begin{array}{cc}\hfill {b}_{1}\hfill & \hfill {b}_{2}\hfill \\ \hfill {b}_{3}\hfill & \hfill {b}_{4}\hfill \end{array}\right),{b}_{j}\in ℝ,{b}_{2}\ne {b}_{3}.$

Here, ${N}_{2}\left(\mathrm{exp}\left(\theta \sqrt{-1}\right),B\right)$ is non-trivial if $\left({b}_{2}-{b}_{3}\right)\mathrm{sin}\theta <0$ and trivial if $\left({b}_{2}-{b}_{3}\right)\mathrm{sin}\theta >0$.

As in [23], the $\diamond$-sum (direct sum) of any two real matrices is defined by

${\left(\begin{array}{cc}\hfill {A}_{1}\hfill & \hfill {B}_{1}\hfill \\ \hfill {C}_{1}\hfill & \hfill {D}_{1}\hfill \end{array}\right)}_{2i×2i}\diamond {\left(\begin{array}{cc}\hfill {A}_{2}\hfill & \hfill {B}_{2}\hfill \\ \hfill {C}_{2}\hfill & \hfill {D}_{2}\hfill \end{array}\right)}_{2j×2j}=\left(\begin{array}{cccc}\hfill {A}_{1}\hfill & \hfill 0\hfill & \hfill {B}_{1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {A}_{2}\hfill & \hfill 0\hfill & \hfill {B}_{2}\hfill \\ \hfill {C}_{1}\hfill & \hfill 0\hfill & \hfill {D}_{1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {C}_{2}\hfill & \hfill 0\hfill & \hfill {D}_{2}\hfill \end{array}\right).$

For every $M\in Sp\left(2n\right)$, the homotopy set $\mathrm{\Omega }\left(M\right)$ of M in $Sp\left(2n\right)$ is defined by

$\mathrm{\Omega }\left(M\right)=\left\{N\in Sp\left(2n\right)|\sigma \left(N\right)\cap 𝕌=\sigma \left(M\right)\cap 𝕌\equiv \mathrm{\Gamma }\text{and}{\nu }_{\omega }\left(N\right)={\nu }_{\omega }\left(M\right)\text{for all}\omega \in \mathrm{\Gamma }\right\},$

where $\sigma \left(M\right)$ denotes the spectrum of M, ${\nu }_{\omega }\left(M\right)\equiv {dim}_{ℂ}{\mathrm{ker}}_{ℂ}\left(M-\omega I\right)$ for $\omega \in 𝕌$. The component ${\mathrm{\Omega }}^{0}\left(M\right)$ of P in $Sp\left(2n\right)$ is defined by the path-connected component of $\mathrm{\Omega }\left(M\right)$ containing M.

For Theorem 2.6 below, cf. [22, Theorem 7.8], [23, Theorems 1.2 and 1.3] and also [24, Theorem 1.8.10, Lemma 2.3.5, and Theorem 8.3.1].

For every $P\mathrm{\in }\mathrm{Sp}\mathit{}\mathrm{\left(}\mathrm{2}\mathit{}n\mathrm{-}\mathrm{2}\mathrm{\right)}$, there exists a continuous path $f\mathrm{\in }{\mathrm{\Omega }}^{\mathrm{0}}\mathit{}\mathrm{\left(}P\mathrm{\right)}$ such that $f\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}P$ and

$f\left(1\right)={N}_{1}{\left(1,1\right)}^{\diamond {p}_{-}}\diamond {I}_{2{p}_{0}}\diamond {N}_{1}{\left(1,-1\right)}^{\diamond {p}_{+}}\diamond {N}_{1}{\left(-1,1\right)}^{\diamond {q}_{-}}\diamond \left(-{I}_{2{q}_{0}}\right)\diamond {N}_{1}{\left(-1,-1\right)}^{\diamond {q}_{+}}$$\diamond {N}_{2}\left(\mathrm{exp}\left({\alpha }_{1}\sqrt{-1}\right),{A}_{1}\right)\diamond \mathrm{\cdots }\diamond {N}_{2}\left(\mathrm{exp}\left({\alpha }_{{r}_{\ast }}\sqrt{-1}\right),{A}_{{r}_{\ast }}\right)$$\diamond {N}_{2}\left(\mathrm{exp}\left({\beta }_{1}\sqrt{-1}\right),{B}_{1}\right)\diamond \mathrm{\cdots }\diamond {N}_{2}\left(\mathrm{exp}\left({\beta }_{{r}_{0}}\sqrt{-1}\right),{B}_{{r}_{0}}\right)$$\diamond R\left({\theta }_{1}\right)\diamond \mathrm{\cdots }\diamond R\left({\theta }_{{r}^{\prime }}\right)\diamond R\left({\theta }_{{r}^{\prime }+1}\right)\diamond \mathrm{\cdots }\diamond R\left({\theta }_{r}\right)\diamond H{\left(±2\right)}^{\diamond h},$(2.8)

where ${\theta }_{j}\mathrm{/}\mathrm{2}\mathit{}\pi \mathrm{\in }ℚ\mathrm{\cap }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ for $\mathrm{1}\mathrm{\le }j\mathrm{\le }{r}^{\mathrm{\prime }}$ and ${\theta }_{j}\mathrm{/}\mathrm{2}\mathit{}\pi \mathrm{\notin }ℚ\mathrm{\cap }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ for ${r}^{\mathrm{\prime }}\mathrm{+}\mathrm{1}\mathrm{\le }j\mathrm{\le }r$. The terms ${N}_{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{exp}\mathit{}\mathrm{\left(}{\alpha }_{j}\mathit{}\sqrt{\mathrm{-}\mathrm{1}}\mathrm{\right)}\mathrm{,}{A}_{j}\mathrm{\right)}$ are non-trivial and ${N}_{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{exp}\mathit{}\mathrm{\left(}{\beta }_{j}\mathit{}\sqrt{\mathrm{-}\mathrm{1}}\mathrm{\right)}\mathrm{,}{B}_{j}\mathrm{\right)}$ are trivial, and the non-negative integers ${p}_{\mathrm{-}}\mathrm{,}{p}_{\mathrm{0}}\mathrm{,}{p}_{\mathrm{+}}\mathrm{,}{q}_{\mathrm{-}}\mathrm{,}{q}_{\mathrm{0}}\mathrm{,}{q}_{\mathrm{+}}\mathrm{,}r\mathrm{,}{r}_{\mathrm{\ast }}\mathrm{,}{r}_{\mathrm{0}}\mathrm{,}h$ satisfy the equality

${p}_{-}+{p}_{0}+{p}_{+}+{q}_{-}+{q}_{0}+{q}_{+}+r+2{r}_{\ast }+2{r}_{0}+h=n-1.$(2.9)

Let

$\gamma \in {\mathcal{𝒫}}_{\tau }\left(2n-2\right)=\left\{\gamma \in C\left(\left[0,\tau \right],Sp\left(2n-2\right)\right)|\gamma \left(0\right)=I\right\}$

and denote the basic normal form decomposition of $P\mathrm{\equiv }\gamma \mathit{}\mathrm{\left(}\tau \mathrm{\right)}$ by (2.8). Then, we have

$i\left({\gamma }^{m}\right)=m\left(i\left(\gamma \right)+{p}_{-}+{p}_{0}-r\right)+2\sum _{j=1}^{r}E\left(\frac{m{\theta }_{j}}{2\pi }\right)-r$$-{p}_{-}-{p}_{0}-\frac{1+{\left(-1\right)}^{m}}{2}\left({q}_{0}+{q}_{+}\right)+2\sum _{j=1}^{{r}_{\ast }}\phi \left(\frac{m{\alpha }_{j}}{2\pi }\right)-2{r}_{\ast },$(2.10)$\nu \left({\gamma }^{m}\right)=\nu \left(\gamma \right)+\frac{1+{\left(-1\right)}^{m}}{2}\left({q}_{-}+2{q}_{0}+{q}_{+}\right)+2\varsigma \left(m,\gamma \left(\tau \right)\right),$(2.11)

where

$\varsigma \left(m,\gamma \left(\tau \right)\right)=r-\sum _{j=1}^{r}\phi \left(\frac{m{\theta }_{j}}{2\pi }\right)+{r}_{\ast }-\sum _{j=1}^{{r}_{\ast }}\phi \left(\frac{m{\alpha }_{j}}{2\pi }\right)+{r}_{0}-\sum _{j=1}^{{r}_{0}}\phi \left(\frac{m{\beta }_{j}}{2\pi }\right).$

We have that $i\mathit{}\mathrm{\left(}\gamma \mathrm{,}\mathrm{1}\mathrm{\right)}$ is odd if $f\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{\right)}\mathrm{=}{N}_{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{\right)}$, ${I}_{\mathrm{2}}$, ${N}_{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{1}\mathrm{\right)}$, $\mathrm{-}{I}_{\mathrm{2}}$, ${N}_{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{-}\mathrm{1}\mathrm{,}\mathrm{-}\mathrm{1}\mathrm{\right)}$ and $R\mathit{}\mathrm{\left(}\theta \mathrm{\right)}$; $i\mathit{}\mathrm{\left(}\gamma \mathrm{,}\mathrm{1}\mathrm{\right)}$ is even if $f\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{\right)}\mathrm{=}$ ${N}_{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{,}\mathrm{-}\mathrm{1}\mathrm{\right)}$ and ${N}_{\mathrm{2}}\mathit{}\mathrm{\left(}\omega \mathrm{,}b\mathrm{\right)}$; $i\mathit{}\mathrm{\left(}\gamma \mathrm{,}\mathrm{1}\mathrm{\right)}$ can be any integer if $\sigma \mathit{}\mathrm{\left(}f\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{\right)}\mathrm{\right)}\mathrm{\cap }𝕌\mathrm{=}\mathrm{\varnothing }$.

The following is the common index jump theorem of Long and Zhu [27] (cf. [27, Theorems 4.1–4.3] and [24]).

Let ${\gamma }_{k}$, $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}q$, be a finite collection of symplectic paths and ${M}_{k}\mathrm{=}{\gamma }_{k}\mathit{}\mathrm{\left(}{\tau }_{k}\mathrm{\right)}\mathrm{\in }S\mathit{}p\mathit{}\mathrm{\left(}\mathrm{2}\mathit{}n\mathrm{-}\mathrm{2}\mathrm{\right)}$. Suppose $\stackrel{\mathrm{^}}{ı}\mathit{}\mathrm{\left(}{\gamma }_{k}\mathrm{,}\mathrm{1}\mathrm{\right)}\mathrm{>}\mathrm{0}$ for all $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}q$. Then, for every $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}q$, there exist infinitely many $\mathrm{\left(}N\mathrm{,}{m}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{m}_{q}\mathrm{\right)}\mathrm{\in }$ ${ℕ}^{q\mathrm{+}\mathrm{1}}$ such that

$\nu \left({\gamma }_{k},2{m}_{k}-1\right)=\nu \left({\gamma }_{k},1\right),$$\nu \left({\gamma }_{k},2{m}_{k}+1\right)=\nu \left({\gamma }_{k},1\right),$$i\left({\gamma }_{k},2{m}_{k}-1\right)+\nu \left({\gamma }_{k},2{m}_{k}-1\right)=2N-\left(i\left({\gamma }_{k},1\right)+2{S}_{{M}_{k}}^{+}\left(1\right)-\nu \left({\gamma }_{k},1\right)\right),$$i\left({\gamma }_{k},2{m}_{k}+1\right)=2N+i\left({\gamma }_{k},1\right),$$i\left({\gamma }_{k},2{m}_{k}\right)\ge 2N-\frac{e\left({M}_{k}\right)}{2},$$i\left({\gamma }_{k},2{m}_{k}\right)+\nu \left({\gamma }_{k},2{m}_{k}\right)\le 2N+\frac{e\left({M}_{k}\right)}{2},$

where ${S}_{{M}_{k}}^{\mathrm{+}}\mathit{}\mathrm{\left(}\mathrm{1}\mathrm{\right)}$ is the splitting number of ${M}_{k}$.

More precisely, by [27, (4.10) and (4.40)] , we have

${m}_{k}=\left(\left[\frac{N}{M\stackrel{^}{ı}\left({\gamma }_{k},1\right)}\right]+{\chi }_{k}\right)M,1\le k\le q,$(2.12)

where ${\chi }_{k}\mathrm{=}\mathrm{0}$ or 1 for $\mathrm{1}\mathrm{\le }k\mathrm{\le }q$ and ${m}_{k}\mathit{}\theta \mathrm{/}\pi \mathrm{\in }ℤ$ whenever $\mathrm{exp}\mathit{}\mathrm{\left(}\theta \mathit{}\sqrt{\mathrm{-}\mathrm{1}}\mathrm{\right)}\mathrm{\in }\sigma \mathit{}\mathrm{\left(}{M}_{k}\mathrm{\right)}$ and $\theta \mathrm{/}\pi \mathrm{\in }ℚ$ for some $\mathrm{1}\mathrm{\le }k\mathrm{\le }q$. Furthermore, given ${M}_{\mathrm{0}}\mathrm{\in }ℕ$, by the proof of [27, Theorem 4.1], we may further require ${M}_{\mathrm{0}}\mathrm{|}N$ (since the closure of the set $\mathrm{\left\{}\mathrm{\left\{}Nv\mathrm{\right\}}\mathrm{\mid }N\mathrm{\in }ℕ\mathrm{,}{M}_{\mathrm{0}}\mathrm{|}N\mathrm{\right\}}$ is still a closed additive subgroup of ${𝕋}^{h}$ for some $h\mathrm{\in }ℕ$, where we use notation as in [27, (4.21)]. Then, we can use the proof of [27, Theorem 4.1, Step 2] to get N).

We also have the following properties in the index iteration theory (cf. [27, Theorem 2.2] or [24, Theorem 10.2.3]).

Let $\gamma \mathrm{\in }{\mathcal{𝒫}}_{\tau }\mathit{}\mathrm{\left(}\mathrm{2}\mathit{}n\mathrm{\right)}$. Then, for any $m\mathrm{\in }ℕ$, there holds

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

where $e\mathit{}\mathrm{\left(}M\mathrm{\right)}$ is the elliptic height defined in Section 1.

3 Proof of Theorem 1.1

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}\left({S}^{3},F\right)\text{with reversibility}\lambda$$\text{and flag curvature}K\text{satisfying}\left(9/4\right){\left(\lambda /\left(1+\lambda \right)\right)}^{2}

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

We have $i\mathit{}\mathrm{\left(}{c}_{k}\mathrm{\right)}\mathrm{\ge }\mathrm{2}$ and $\stackrel{\mathrm{^}}{ı}\mathit{}\mathrm{\left(}{c}_{k}\mathrm{\right)}\mathrm{>}\mathrm{3}$ for $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}q$.

Proof.

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

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

and

$\stackrel{^}{ı}\left({c}_{k}\right)\ge 2\sqrt{\delta }\frac{\lambda +1}{\lambda }>3.$

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

Combining Lemma 3.1 with Theorem 2.8, it follows that

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

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

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

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

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

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

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

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

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

$i\left({c}_{k}^{2{m}_{k}}\right)=2N-{S}_{{M}_{k}}^{+}\left(1\right)-C\left({M}_{k}\right)+2{\mathrm{\Delta }}_{k},$(3.10)$i\left({c}_{k}^{2{m}_{k}}\right)+\nu \left({c}_{k}^{2{m}_{k}}\right)=2N+{p}_{{k}_{0}}+{p}_{{k}_{+}}+{q}_{{k}_{-}}+{q}_{{k}_{0}}$$+2{r}_{{k}_{0}}^{\prime }-2\left({r}_{{k}_{\ast }}-{r}_{{k}_{\ast }}^{\prime }\right)+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 $R\left(\theta \right)$, ${N}_{2}\left(\mathrm{exp}\left(\alpha \sqrt{-1}\right),A\right)$, and ${N}_{2}\left(\mathrm{exp}\left(\beta \sqrt{-1}\right),B\right)$ 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 $R\left(\theta \right)$, ${N}_{2}\left(\mathrm{exp}\left(\alpha \sqrt{-1}\right),A\right)$, and ${N}_{2}\left(\mathrm{exp}\left(\beta \sqrt{-1}\right),B\right)$ 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

${\mathrm{\Delta }}_{k}\equiv \sum _{0<\left\{{m}_{k}\theta /\pi \right\}<\delta }{S}_{{M}_{k}}^{-}\left(\mathrm{exp}\left(\theta \sqrt{-1}\right)\right)\le {r}_{k}-{r}_{k}^{\prime }+{r}_{{k}_{\ast }}-{r}_{{k}_{\ast }}^{\prime },C\left({M}_{k}\right)\equiv \sum _{\theta \in \left(0,2\pi \right)}{S}_{{M}_{k}}^{-}\left(\mathrm{exp}\left(\theta \sqrt{-1}\right)\right),$(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 $\left({S}^{3},F\right)$ whose flag curvature satisfies ${\left(\lambda /\left(1+\lambda \right)\right)}^{2}. 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 $\mathrm{\left(}{S}^{\mathrm{3}}\mathrm{,}F\mathrm{\right)}$ whose flag curvature satisfies ${\mathrm{\left(}\lambda \mathrm{/}\mathrm{\left(}\mathrm{1}\mathrm{+}\lambda \mathrm{\right)}\mathrm{\right)}}^{\mathrm{2}}\mathrm{<}K\mathrm{\le }\mathrm{1}$. Moreover, there exist infinitely many pairs of $\mathrm{\left(}q\mathrm{+}\mathrm{1}\mathrm{\right)}$-tuples of the form $\mathrm{\left(}N\mathrm{,}{m}_{\mathrm{1}}\mathrm{,}{m}_{\mathrm{2}}\mathrm{,}\mathrm{\dots }\mathrm{,}{m}_{q}\mathrm{\right)}\mathrm{\in }{ℕ}^{q\mathrm{+}\mathrm{1}}$ and $\mathrm{\left(}{N}^{\mathrm{\prime }}\mathrm{,}{m}_{\mathrm{1}}^{\mathrm{\prime }}\mathrm{,}{m}_{\mathrm{2}}^{\mathrm{\prime }}\mathrm{,}\mathrm{\dots }\mathrm{,}{m}_{q}^{\mathrm{\prime }}\mathrm{\right)}\mathrm{\in }{ℕ}^{q\mathrm{+}\mathrm{1}}$ such that

$i\left({c}_{1}^{2{m}_{1}}\right)+\nu \left({c}_{1}^{2{m}_{1}}\right)=2N+2,$$\mathrm{ }{\overline{C}}_{2N+2}\left(E,{c}_{1}^{2{m}_{1}}\right)=ℚ,$(3.13)$i\left({c}_{2}^{2{m}_{2}^{\prime }}\right)+\nu \left({c}_{2}^{2{m}_{2}^{\prime }}\right)=2{N}^{\prime }+2,$${\overline{C}}_{2{N}^{\prime }+2}\left(E,{c}_{2}^{2{m}_{2}^{\prime }}\right)=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

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

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

for all $\mathrm{0}\mathrm{\le }j\mathrm{<}\nu \mathit{}\mathrm{\left(}{c}_{k}^{n\mathit{}\mathrm{\left(}{c}_{k}\mathrm{\right)}}\mathrm{\right)}$, $k\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}$, and then $\stackrel{\mathrm{^}}{\chi }\mathit{}\mathrm{\left(}{c}_{k}\mathrm{\right)}\mathrm{\le }\mathrm{1}$ 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}\left(E,{c}_{1}^{2{m}_{1}}\right)$$=dim{H}_{2N+2-i\left({c}_{1}^{2{m}_{1}}\right)}{\left({N}_{{c}_{1}^{2{m}_{1}}}\cup \left\{{c}_{1}^{2{m}_{1}}\right\},{N}_{{c}_{1}^{2{m}_{1}}}\right)}^{ϵ\left({c}_{1}^{2{m}_{1}}\right){ℤ}_{2{m}_{1}}}$$=dim{H}_{\nu \left({c}_{1}^{2{m}_{1}}\right)}{\left({N}_{{c}_{1}^{2{m}_{1}}}\cup \left\{{c}_{1}^{2{m}_{1}}\right\},{N}_{{c}_{1}^{2{m}_{1}}}\right)}^{ϵ\left({c}_{1}^{2{m}_{1}}\right){ℤ}_{2{m}_{1}}}$$={k}_{\nu \left({c}_{1}^{2{m}_{1}}\right)}^{ϵ\left({c}_{1}^{2{m}_{1}}\right)}\left({c}_{1}^{2{m}_{1}}\right),$

which implies that

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

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

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

for any $0\le j\le \nu \left({c}_{1}^{2{m}_{1}}\right)$ 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 $R\left({\theta }_{1}\right)\diamond R\left({\theta }_{2}\right)$ or $R\left({\theta }_{1}\right)\diamond {N}_{1}\left(\lambda ,b\right)$ for some ${\theta }_{1}/2\pi \in \left(0,1\right)\ℚ$, $\lambda =±1$, and $b=0,1$. Then,

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

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

Then, (3.20) yields

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

for all $0 and for $1\le m, which together with (2.3) and (3.19) gives

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

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}$$\text{in Lemmas 3.2 and 3.3 and a third closed geodesic}{c}_{3}\text{on}\left({S}^{3},F\right)\text{with reversibility}\lambda \text{and}$$\text{flag curvature}K\text{satisfying}\left(9/4\right){\left(\lambda /\left(1+\lambda \right)\right)}^{2}

${c}_{1}^{m}$ 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 $m\mathrm{\in }ℕ$, ${c}_{\mathrm{2}}^{m}$ 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 $R\left({\theta }_{1}\right)\diamond R\left({\theta }_{2}\right)$ or $R\left({\theta }_{1}\right)\diamond {N}_{1}\left(\lambda ,b\right)$ for some ${\theta }_{1}/2\pi \in \left(0,1\right)\ℚ$, $\lambda =±1$, and $b=0,1$. Combining this fact with Lemma 3.1 and (3.7), we have

$i\left({c}_{k}^{m}\right)+\nu \left({c}_{k}^{m}\right)\le 2N-i\left({c}_{k}\right)-{p}_{{k}_{-}}+{p}_{{k}_{+}}\le 2N-1$(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 $R\left({\theta }_{1}\right)\diamond {N}_{1}\left(1,-1\right)$ and $i\left({c}_{k}\right)=2$, but $i\left({c}_{k}\right)\in 2ℕ-1$ when ${P}_{{c}_{k}}$ is conjugate to $R\left({\theta }_{1}\right)\diamond {N}_{1}\left(1,-1\right)$, thus the equality in (3.21) does not hold. Then,

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

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

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

$i\left({c}_{2}^{2{m}_{2}^{\prime }}\right)=i\left({c}_{2}^{2{m}_{2}}\right)\left(\mathrm{mod} 2\right)$

by (2.10) of Theorem 2.6. So, $i\left({c}_{2}^{2{m}_{2}}\right)+\nu \left({c}_{2}^{2{m}_{2}}\right)$ is even since $i\left({c}_{2}^{2{m}_{2}^{\prime }}\right)+\nu \left({c}_{2}^{2{m}_{2}^{\prime }}\right)$ 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 $\nu \left({c}_{3}^{2{m}_{3}-1}\right)=\nu \left({c}_{3}\right)$, we have

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

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.,

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

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

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

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

We continue the proof by distinguishing two cases.

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

$i\left({c}_{3}^{m}\right)+\nu \left({c}_{3}^{m}\right)-2=i\left({c}_{3}^{m}\right)=mi\left({c}_{3}\right)=m\left(i\left({c}_{3}\right)+\nu \left({c}_{3}\right)-2\right)=2m$(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 $m\in ℕ$, i.e.,

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

Thus, ${c}_{3}^{2{m}_{3}-1}$ has contribution to ${M}_{2N}$ and ${k}_{\nu \left({c}_{3}\right)}\left({c}_{3}\right)=1$, since otherwise ${c}_{3}^{2{m}_{3}-1}$ contributes to ${M}_{2N-1}$ and ${k}_{1}\left({c}_{3}\right)\ne 0$, and then ${c}_{3}^{m}$ has no contribution to ${M}_{2N}$ for any $m\in ℕ$ by (3.25), which contradicts (3.26). Now, ${k}_{\nu \left({c}_{3}\right)}\left({c}_{3}\right)=1$ 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. $i\mathit{}\mathrm{\left(}{c}_{\mathrm{3}}^{\mathrm{2}\mathit{}{m}_{\mathrm{3}}\mathrm{-}\mathrm{1}}\mathrm{\right)}\mathrm{+}\nu \mathit{}\mathrm{\left(}{c}_{\mathrm{3}}^{\mathrm{2}\mathit{}{m}_{\mathrm{3}}\mathrm{-}\mathrm{1}}\mathrm{\right)}\mathrm{=}\mathrm{2}\mathit{}N\mathrm{-}\mathrm{1}$. In this case, by (3.24), one of the following cases may happen.

• (i)

$i\left({c}_{3}\right)=3$ and ${p}_{{3}_{+}}=2$.

• (ii)

$i\left({c}_{3}\right)=2$ and ${p}_{{3}_{+}}=1$.

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

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

$i\left({c}_{3}^{m}\right)+\nu \left({c}_{3}^{m}\right)=mi\left({c}_{3}\right)+\nu \left({c}_{3}^{m}\right)=2m+1$(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 ${k}_{\nu \left({c}_{3}\right)}\left({c}_{3}\right)=1$, which together with (3.27) implies that ${c}_{3}^{m}$ has no contribution to ${M}_{2N}$ for any $m\in ℕ$, 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

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

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

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

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

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

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

Now, we can obtain that

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

for all $1\le m.

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

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

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

$\sum _{j=1}^{3}\frac{\stackrel{^}{\chi }\left({c}_{j}\right)}{\stackrel{^}{ı}\left({c}_{j}\right)}<\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1,$

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

$2N\ge i\left({c}_{2}^{2{m}_{2}}\right)+\nu \left({c}_{2}^{2{m}_{2}}\right)$(3.33)$=2N+\left({p}_{{2}_{0}}+{p}_{2+}+{q}_{{2}_{-}}+{q}_{{2}_{0}}+2{r}_{{2}_{0}}^{\prime }+{r}_{2}^{\prime }\right)-\left({r}_{2}-{r}_{2}^{\prime }\right)$$\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 $i\left({c}_{2}^{2{m}_{2}}\right)+\nu \left({c}_{2}^{2{m}_{2}}\right)\ne 2N$ and by (3.14), we have that

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

is even since it has the same parity with $i\left({c}_{2}^{2{m}_{2}^{\prime }}\right)+\nu \left({c}_{2}^{2{m}_{2}^{\prime }}\right)$. Thus, by (3.33), we obtain $i\left({c}_{2}^{2{m}_{2}}\right)+\nu \left({c}_{2}^{2{m}_{2}}\right)\le 2N-2$, 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 ${m}_{0}\in ℕ$, then 1 must be an eigenvalue of ${P}_{{c}_{3}^{2l{m}_{0}}}$ for any $l\in ℕ$ 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 $m\in ℕ$ 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. ∎

The authors would like to sincerely thank Professor Yiming Long for his valuable help and his encouragement. The authors also thank him sincerely for his comments, suggestions, and helpful discussions about the closed geodesic problem. Finally, the authors sincerely thank the referee for her/his careful reading, valuable comments, and suggestions on this paper.

References

• [1]

Anosov D. V., Geodesics in Finsler geometry (in Russian), Proceedings of the International Congress of Mathematicians (Vancouver 1974), Vol. 2, Canad. Math. Congress, Montreal (1975), 293–297; translation in Amer. Math. Soc. Transl. Ser. 2 109 (1977), 81–85.  Google Scholar

• [2]

Ballmann W., Thobergsson G. and Ziller W., Closed geodesics on positively curved manifolds, Ann. of Math. (2) 116 (1982), no. 2, 213–247.  Google Scholar

• [3]

Ballmann W., Thobergsson G. and Ziller W., Existence of closed geodesics on positively curved manifolds, J. Differential Geom. 18 (1983), no. 2, 221–252.  Google Scholar

• [4]

Bangert V., On the existence of closed geodesics on two-spheres, Internat. J. Math. 4 (1993), no. 1, 1–10.  Google Scholar

• [5]

Bangert V. and Long Y., The existence of two closed geodesics on every Finsler 2-sphere, Math. Ann. 346 (2010), no. 2, 335–366.  Google Scholar

• [6]

Bott R., On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956), 171–206.  Google Scholar

• [7]

Chang K. C., Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser, Boston, 1993.  Google Scholar

• [8]

Duan H., Non-hyperbolic closed geodesics on positively curved Finsler spheres, J. Funct. Anal. 269 (2015), no. 11, 3645–3662.  Google Scholar

• [9]

Duan H., Two elliptic closed geodesics on positively curved Finsler spheres, preprint 2015, http://arxiv.org/abs/1504.00245.  Google Scholar

• [10]

Duan H. and Long Y., Multiple closed geodesics on bumpy Finsler n-spheres, J. Differential Equations 233 (2007), no. 1, 221–240.  Google Scholar

• [11]

Duan H. and Long Y., Multiplicity and stability of closed geodesics on bumpy Finsler 3-spheres, Calc. Var. Partial Differential Equations 31 (2008), no. 4, 483–496.  Google Scholar

• [12]

Duan H. and Long Y., The index growth and mutiplicity of closed geodesics, J. Funct. Anal. 259 (2010), no. 7, 1850–1913.  Google Scholar

• [13]

Duan H., Long Y. and Wang W., The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds, preprint 2015, http://arxiv.org/abs/1510.02872.  Google Scholar

• [14]

Duan H., Long Y. and Wang W., Two closed geodesics on compact simply-connected bumpy Finsler manifolds, J. Differential Geom., to appear.  Google Scholar

• [15]

Franks J., Geodesics on ${S}^{2}$ and periodic points of annulus diffeomorphisms, Invent. Math. 108 (1992), no. 2, 403–418.  Google Scholar

• [16]

Hingston N., Equivariant Morse theory and closed geodesics, J. Differential Geom. 19 (1984), no. 1, 85–116.  Google Scholar

• [17]

Hingston N., On the growth of the number of closed geodesics on the two-sphere, Int. Math. Res. Not. IMRN 9 (1993), no. 9, 253–262.  Google Scholar

• [18]

Hingston N., On the length of closed geodesics on a two-sphere, Proc. Amer. Math. Soc. 125 (1997), no. 10, 3099–3106.  Google Scholar

• [19]

Katok A. B., Ergodic perturbations of degenerate integrable Hamiltonian systems (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 539–576; translation in Math. USSR-Izv. 7 (1973), 535–571.  Google Scholar

• [20]

Klingenberg W., Lectures on Closed Geodesics, Grundlehren Math. Wiss. 230, Springer, Berlin, 1978.  Google Scholar

• [21]

Liu C. G., The relation of the Morse index of closed geodesics with the Maslov-type index of symplectic paths, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 2, 237–248.  Google Scholar

• [22]

Long Y., Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999), no. 1, 113–149.  Google Scholar

• [23]

Long Y., Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math. 154 (2000), no. 1, 76–131.  Google Scholar

• [24]

Long Y., Index Theory for Symplectic Paths with Applications, Progr. Math. 207, Birkhäuser, Basel, 2002.  Google Scholar

• [25]

Long Y., Multiplicity and stability of closed geodesics on Finsler 2-spheres, J. Eur. Math. Soc. (JEMS) 8 (2006), no. 2, 341–353.  Google Scholar

• [26]

Long Y. and Duan H., Multiple closed geodesics on 3-spheres, Adv. Math. 221 (2009), no. 6, 1757–1803.  Google Scholar

• [27]

Long Y. and Zhu C., Closed characteristics on compact convex hypersurfaces in ${ℝ}^{2n}$, Ann. of Math. (2) 155 (2002), no. 2, 317–368.  Google Scholar

• [28]

Rademacher H.-B., On the average indices of closed geodesics, J. Differential Geom. 29 (1989), no. 1, 65–83.  Google Scholar

• [29]

Rademacher H.-B., Morse Theorie und geschlossene Geodätische, Bonner Math. Schriften 229, Universität Bonn, Bonn, 1992.  Google Scholar

• [30]

Rademacher H.-B., A sphere theorem for non-reversible Finsler metric, Math. Ann. 328 (2004), no. 3, 373–387.  Google Scholar

• [31]

Rademacher H.-B., Existence of closed geodesics on positively curved Finsler manifolds, Ergodic Theory Dynam. Systems 27 (2007), no. 3, 957–969.  Google Scholar

• [32]

Rademacher H.-B., The second closed geodesic on Finsler spheres of dimension $n>2$, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1413–1421.  Google Scholar

• [33]

Wang W., Closed geodesics on positively curved Finsler spheres, Adv. Math. 218 (2008), no. 5, 1566–1603.  Google Scholar

• [34]

Wang W., Closed geodesics on Finsler spheres, Calc. Var. Partial Differential Equations 45 (2012), no. 1–2, 253–272.  Google Scholar

• [35]

Wang W., On a conjecture of Anosov, Adv. Math. 230 (2012), no. 4–6, 1597–1617.  Google Scholar

• [36]

Wang W., On the average indices of closed geodesics on positively curved Finsler spheres, Math. Ann. 355 (2013), no. 3, 1049–1065.  Google Scholar

• [37]

Ziller W., Geometry of the Katok examples, Ergodic Theory Dynam. Systems 3 (1982), no. 1, 135–157.  Google Scholar

Revised: 2015-08-27

Accepted: 2015-08-28

Published Online: 2016-01-14

Published in Print: 2016-02-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11131004

Award identifier / Grant number: 11471169

Award identifier / Grant number: 11401555

Award identifier / Grant number: 11371339

Funding Source: China Postdoctoral Science Foundation

Award identifier / Grant number: 2014T70589

The first author is partially supported by the NSFC (grant nos. 11131004, 11471169), the LPMC of MOE of China, and Nankai University. The second author is partially supported by the NSFC (grant nos. 11401555, 11371339), the China Postdoctoral Science Foundation (grant no. 2014T70589), and the CUSF (grant no. WK3470000001).

Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 1, Pages 159–171, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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