The irrational ratio of average indices of closed geodesics on positively curved Finsler spheres

Linear Algebra Applications Nonlinear Analysisand 128its(2015) 36–47 466 (2015) 102–116

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Linear Algebra and its Applications Nonlinear Analysis www.elsevier.com/locate/laa www.elsevier.com/locate/na

problem of Jacobi matrixon The irrational Inverse ratio ofeigenvalue average indices of closed geodesics withFinsler mixedspheres data positively curved b,∗ 1 Ying Huagui Duan a , Hui LiuWei Department of Mathematics, Nanjing University of Aeronautics Astronautics, School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, PR and China Nanjing 210016, PR China Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR China a

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abstract

Article history: In this paper, the inverse eigenvalue problem of reconstructing Received 16 January 2014 a Jacobi matrix from its eigenvalues, its leading principal Article history: In this that for every Finsler n-sphere (S n , F ) carrying finitely many Accepted 20 September 2014paper, we provesubmatrix and part of the eigenvalues of its submatrix Received 1 January 2015 prime closed Available online 22 October 2014 geodesics for n ≥ 6 with reversibility λ and flag curvature K satisfying considered. The necessary and sufficient conditions for Accepted 24 July 2015 Submitted by Y. Wei( λ )2 < K ≤ 1, thereisexist at least [ n ]−1 prime closed geodesics such that for any λ+1 2 the existence and uniqueness of the solution are derived. Communicated by S. Carl MSC: 53C22 53C60 58E10 Keywords: Finsler spheres Closed geodesics Index iteration Average index Irrational ratio

MSC: 15A18 15A57

two elements among them, the ratio of their average indices is an irrational number. Furthermore, a numerical algorithm and some numerical In addition, if the metric is bumpy, then there exist at least n − 2 closed geodesics examples are given. satisfying the above property. © 2014 Published by Elsevier Inc. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Jacobi matrix Eigenvalue Inverse problem Submatrix

1. Introduction and main results In this paper, we study some properties of closed geodesics on positively curved Finsler spheres. 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. As usual, on any Finsler n-sphere S n = (S n , F ), a closed geodesic c : S 1 = R/Z → S n is prime if it is not a multiple covering (i.e., iteration) of any other closed geodesics. Here the mth iteration cm of c is defined by cm (t) = c(mt). The inverse curve c−1 of c is defined by c−1 (t) = c(1−t) address: [email protected]. for t ∈ R. Note that onE-mail an irreversible Finsler manifold, the inverse curve of a closed geodesic is not a closed 1 Tel.: +86 13914485239. geodesic in general. We call two prime closed geodesics c and d distinct if there is no θ ∈ (0, 1) such that c(t) = d(t + θ) for http://dx.doi.org/10.1016/j.laa.2014.09.031 all t ∈ R. We shall omit the word distinct when we talk about more than one prime 0024-3795/© 2014 Published Elsevier Inc. n-sphere, two closed geodesics c and d are called closed geodesic. On a reversible Finsler (or by Riemannian) ∗ Corresponding author. E-mail addresses: [email protected] (H. Duan), [email protected] (H. Liu).

http://dx.doi.org/10.1016/j.na.2015.07.025 0362-546X/© 2015 Elsevier Ltd. All rights reserved.

H. Duan, H. Liu / Nonlinear Analysis 128 (2015) 36–47

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geometrically distinct if c(S 1 ) ̸= d(S 1 ), i.e., their image sets in S n are distinct. We denote the set of all distinct prime closed geodesics on a Finsler manifold by CG(M, F ). For a closed geodesic c on (S n , F ), denote by Pc the linearized Poincar´e map of c. Then Pc ∈ Sp(2n − 2) is symplectic. For any M ∈ Sp(2k), we define the elliptic height e(M ) of M to be the total algebraic multiplicity of all eigenvalues of M on the unit circle U = {z ∈ C| |z| = 1} in the complex plane C. Since M is symplectic, e(M ) is even and 0 ≤ e(M ) ≤ 2k. A closed geodesic c is called non-degenerate if 1 is not 0 an eigenvalue of Pc ; irrationally elliptic if, in the homotopy component  Ω (Pc ) ofPc (cf. Section 3 below), Pc can be connected to the -product of (n − 1) rotation matrices

cos θi sin θi

− sin θi cos θi

with θi being irrational

multiples of π for 1 ≤ i ≤ n − 1. A Finsler sphere (S n , F ) is called bumpy if all the closed geodesics on it are non-degenerate. Following Rademacher in [18], the reversibility λ = λ(M, F ) of a compact Finsler manifold (M, F ) is defined to be λ := max{F (−X) | X ∈ T M, F (X) = 1} ≥ 1. It was quite surprising when Katok [7] in 1973 found some irreversible Finsler metrics on CROSS with only finitely many prime closed geodesics and all closed geodesics are non-degenerate and elliptic. The smallest number of closed geodesics on S n that one obtains in these examples is 2[ n+1 2 ] (cf. [24]). Then Anosov in I.C.M. of 1974 conjectured that the lower bound of the number of closed geodesics on any Finsler sphere (S n , F ) should be 2[ n+1 2 ], i.e., the number of closed geodesics in Katok’s example. In [19], Rademacher studied the existence and stability of closed geodesics on positively curved Finsler manifolds. In particular, he proved that there are always n2 − 1 prime closed geodesics of length ≤ 2nπ on every Finsler n-sphere (S n , F )  2 λ satisfying λ+1 < K ≤ 1. In [1] of Bangert and Long, they proved that on any Finsler 2-sphere (S 2 , F ),

there exist at least two prime closed geodesics, which answers Anosov’s conjecture for S 2 . In [14] of Long and Wang, they further proved the existence of at least two irrationally elliptic prime closed geodesics on every Finsler 2-sphere (S 2 , F ) provided # CG(S 2 , F ) < +∞. In [21], Wang proved that there are always three prime  2 λ closed geodesics on every Finsler 3-sphere (S 3 , F ) satisfying λ+1 < K ≤ 1. In a series papers [13,4,5] of Long and Duan, they obtained some multiplicity results on compact simply connected Finsler manifolds. In [20], Rademacher proved there exist at least two prime closed geodesics on any bumpy n-sphere. In [22],  2 λ Wang proved Anosov’s conjecture for bumpy n-spheres satisfying λ+1 < K ≤ 1. In [23], Wang considered the number of closed geodesics on positively curved Finsler spheres which have irrational average indices. Motivated by the above results, we prove the following results in this paper: Theorem 1.1. On every Finsler n-sphere (S n , F ) satisfying # CG(S n , F ) < +∞ for n ≥ 6 with reversibility  2 λ λ and flag curvature K satisfying λ+1 < K ≤ 1, there exist at least [ n2 ] − 1 prime closed geodesics such that for any two elements among them, the ratio of their average indices is an irrational number. Theorem 1.2. On every bumpy Finsler n-sphere (S n , F ) satisfying # CG(S n , F ) < +∞ for n ≥ 6 with  2 λ reversibility λ and flag curvature K satisfying λ+1 < K ≤ 1, there exist at least n − 2 prime closed geodesics such that for any two elements among them, the ratio of their average indices is an irrational number. Remark 1.3. Note that the Katok metric on S n has exactly 2[ n+1 2 ] prime closed geodesics and the ratio of average indices of any two of them is irrational. In fact, by P. 145 of [24], we know that all these closed geodesics are irrationally elliptic, which, together with the proof of Theorem 1.2 of [3] (cf. Claims 1–3 in Section 3 of [3]), yields that these closed geodesics are infinitely variationally visible (defined by Definition 2.6

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below). Then by (2.10) below we have ˆi(c1 ) ˆi(c2 ) = L(c1 ) L(c2 )

(1.1)

for any two prime closed geodesics c1 , c2 found in the Katok’s example, where L(ck ) denotes the length of the closed geodesic ck , k = 1, 2. On the other hand, by P. 142 of [24], the lengths of these closed geodesics are 2π/(1 + pα/pj ) and 2π/(1 − pα/pj ) respectively, where α ∈ (0, 1)\Q is small enough, and the n+1 pj ∈ Z, j = 1, . . . , [ n+1 2 ] are relatively prime, and p = p1 · · · p[ 2 ] as denoted on P. 139 of [24]. So, by (1.1), the ratio of average indices of any two distinct ones among these closed geodesics has the following forms (1 ≤ j, k ≤ [ n+1 2 ]): 1 + pα/pj 1 + pα/pk

(j ̸= k),

1 − pα/pj 1 − pα/pk

(j ̸= k),

1 + pα/pj , 1 − pα/pk

(1.2)

which are irrational since α is irrational. Remark 1.4. By Theorems 1.1–1.2, for Finsler n-sphere (S n , F ) satisfying # CG(S n , F ) < +∞ with 2  λ < K ≤ 1, it seems that all the Morse indices of reversibility λ and flag curvature K satisfying λ+1 the closed geodesics are similar to those in the Katok’s example. We also note that recently some similar results for closed characteristics on compact convex hypersurfaces in R2n were obtained by Hu and Ou in [6] and our proofs are also motivated by theirs. The main ingredients in our proof are: the common index jump theorem of Long and Zhu (Theorem 4.3 of [15]) with some further discussions and Fadell–Rabinowitz index theory for closed geodesics. In this paper, let N, Z, Q, Q+ , R, and C denote the sets of natural integers, integers, rational numbers, positive 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 X the quotient space X/S 1 . We define the functions [a] = max{k ∈ Z | k ≤ a} and {a} = a − [a] for any a ∈ R. 2. Critical point theory for closed geodesics Let M be a compact and simply connected manifold with a Finsler metric F . Closed geodesics are critical  2 points of the energy functional E(γ) = 12 S 1 F (γ(t)) ˙ dt on the Hilbert manifold Λ = ΛM of H 1 -maps  γ : S 1 → M . The length functional L on Λ is defined by L(γ) = S 1 F (γ(t))dt. ˙ An S 1 action is defined by 1 (s · γ)(t) = γ(t + s) for all γ ∈ Λ and s, t ∈ S . The index form of the functional E is well defined along any closed geodesic c on M , which we denote by E ′′ (c). As usual we denote by i(c) and ν(c) the Morse index and nullity of E at c. For κ ∈ R we denote by Λκ = {d ∈ Λ | E(d) ≤ κ}. For a closed geodesic c, denote by cm the m-fold iteration of c and Λ(cm ) = {γ ∈ Λ | E(γ) < E(cm )}. Recall that respectively the mean index ˆi(c) and the S 1 -critical modules of cm are defined by m ˆi(c) = lim i(c ) , m→∞ m

  C ∗ (E, cm ) = H∗ (Λ(cm ) ∪ S 1 · cm )/S 1 , Λ(cm )/S 1 .

(2.1)

We call a closed geodesic satisfying the isolation condition, if the following holds: (Iso) For all m ∈ N the orbit S 1 · cm 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). n If c has multiplicity m, then the subgroup Zm = { m | 0 ≤ n < m} of S 1 acts on C ∗ (E, c). As studied in p. 59 of [16], for all m ∈ N, let H∗ (X, A)±Zm = {[ξ] ∈ H∗ (X, A) | T∗ [ξ] = ±[ξ]}, where T is a generator of the Zm -action. On S 1 -critical modules of cm , the following lemma holds:

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Lemma 2.1 (Cf. Satz 6.11 of [16] or Proposition 3.12 of [1]). Let c be a prime closed geodesic on a Finsler manifold (M, F ) satisfying (Iso). Then there exist two sets Uc−m and Nc−m , the so-called local negative disk and the local characteristic manifold at cm respectively, such that ν(cm ) = dim Nc−m and   C q (E, cm ) ≡ Hq (Λ(cm ) ∪ S 1 · cm )/S 1 , Λ(cm )/S 1 +Zm  . = Hi(cm ) (Uc−m ∪ {cm }, Uc−m ) ⊗ Hq−i(cm ) (Nc−m ∪ {cm }, Nc−m ) (i) When ν(cm ) = 0, there holds  Q, if i(cm ) − i(c) ∈ 2Z, and q = i(cm ) C q (E, c ) = 0, otherwise. m

(ii) When ν(cm ) > 0, there holds C q (E, cm ) = Hq−i(cm ) (Nc−m ∪ {cm }, Nc−m )β(c m

where β(cm ) = (−1)i(c

)−i(c)

m

)Zm

,

.

Let kj (cm ) ≡ dim Hj (Nc−m ∪ {cm }, Nc−m ),

kj±1 (cm ) ≡ dim Hj (Nc−m ∪ {cm }, Nc−m )±Zm .

(2.2)

Then we have Lemma 2.2 (Cf. [16,1]). Let c be a prime closed geodesic on a Finsler manifold (M, F ) satisfying (Iso). (i) There hold 0 ≤ kj±1 (cm ) ≤ kj (cm ), ∀ m ∈ N, j ∈ Z, kj (cm ) = 0 whenever j ̸∈ [0, ν(cm )] and kν(cm ) (cm ) ≤ 1. If kν(cm ) (cm ) = 1, then kj (cm ) = 0 when j ∈ [0, ν(cm )). (ii) Suppose for some integer m = np ≥ 2 with n and p ∈ N the nullities satisfy ν(cm ) = ν(cn ). Then there hold kj (cm ) = kj (cn ) and kj±1 (cm ) = kj±1 (cn ) for any integer j. Next we recall the Fadell–Rabinowitz index in a relative version due to [17]. Let X be an S 1 -space, A ⊂ X a closed S 1 -invariant subset. Note that the cup product defines a homomorphism HS∗ 1 (X) ⊗ HS∗ 1 (X, A) → HS∗ 1 (X, A) : (ζ, z) → ζ ∪ z,

(2.3)

where HS∗ 1 is the S 1 -equivariant cohomology with rational coefficients in the sense of A. Borel (cf. Chapter IV of [2]). We fix a characteristic class η ∈ H 2 (CP ∞ ). Let f ∗ : H ∗ (CP ∞ ) → HS∗ 1 (X) be the homomorphism induced by a classifying map f : XS 1 → CP ∞ . Now for γ ∈ H ∗ (CP ∞ ) and z ∈ HS∗ 1 (X, A), let γ · z = f ∗ (γ) ∪ z. Then the order ord η (z) with respect to η is defined by ord η (z) = inf{k ∈ N ∪ {∞} | η k · z = 0}.

(2.4)

By Proposition 3.1 of [17], there is an element z ∈ HSn+1 (Λ, Λ0 ) of infinite order, i.e., ord η (z) = ∞. For κ ≥ 0, 1 κ 0 0 we denote by jκ : (Λ , Λ ) → (Λ Λ ) the natural inclusion and define the function dz : R≥0 → N ∪ {∞}: dz (κ) = ord η (jκ∗ (z)). Denote by dz (κ−) = limϵ↘0 dz (κ − ϵ), where t ↘ a means t > a and t → a. Then we have the following property.

(2.5)

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Lemma 2.3 (Cf. Section 5 of [17]). The function dz is non-decreasing and limλ↘κ dz (λ) = dz (κ). Each discontinuous point of dz is a critical value of the energy functional E. In particular, if dz (κ) − dz (κ−) ≥ 2, then there are infinitely many prime closed geodesics c with energy κ. For each i ≥ 1, define κi = inf{δ ∈ R | dz (δ) ≥ i}.

(2.6)

Then we have the following. Lemma 2.4 (Cf. Lemma 2.3 of [23]). Suppose there are only finitely many prime closed geodesics on (S n , F ). Then each κi is a critical value of E. If κi = κj for some i < j, then there are infinitely many prime closed geodesics on (S n , F ). Lemma 2.5 (Cf. Lemma 2.4 of [23]). Suppose there are only finitely many prime closed geodesics on (S n , F ). Then for every i ∈ N, there exists a closed geodesic c on (S n , F ) such that C 2i+dim(z)−2 (E, c) ̸= 0,

(2.7)

i(c) ≤ 2i + n − 1 ≤ i(c) + ν(c).

(2.8)

E(c) = κi , where dim(z) = n + 1. Note that by (i) of Lemma 2.2, (2.7) yields E(c) = κi , Similar to Definition 1.4 of [15], we have

Definition 2.6. A prime closed geodesic c is (m, i)-variationally visible, if there exist some m, i ∈ N such that (2.7) holds for cm and κi . We call c infinitely variationally visible, if there exist infinitely many m, i ∈ N such that c is (m, i)-variationally visible. Denote by V(S n , F ) (or V∞ (S n , F )) the set of variationally visible(or infinitely variationally visible) closed geodesics on (S n , F ). The following theorem is crucial for the proof of our main theorems. Theorem 2.7. Suppose there are only finitely many prime closed geodesics on (S n , F ). Then there exists an integer K ≥ 0 and an injection map p : N + K −→ V∞ (S n , F ) × N such that (i) For any i ∈ N + K, c ∈ CG(S n , F ) and m ∈ N satisfying p(i) = (c, m) and (2.8) holds for cm and κi , and (ii) For any ik ∈ N + K, i1 < i2 , ck ∈ CG(S n , F ) satisfying p(ik ) = (ck , mk ) with k = 1, 2, 1 ˆi(cm ˆ m2 1 ) < i(c2 ).

(2.9)

Proof. (i) By Lemma 2.5, for each i ∈ N, there is a prime closed geodesic c such that c is (m, i)-variationally visible for some m ∈ N, i.e., c ∈ V(S n , F ). We define a map p1 : N → V(S n , F ) × N by p1 (i) = (c, m). Since # CG(S n , F ) < +∞, by Lemma 2.3, (2.6) and Lemma 2.4, we have κi < κj whenever i < j. Thus if p1 (i) = p1 (j) = (c, m) for some i < j, by (2.7) we obtain κi = E(cm ) = κj . This contradiction proves that p1 is injective.

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Since # CG(S n , F ) < +∞, there exists an integer K ≥ 0 such that all critical values κi+K with i ∈ N come from iterations of elements in V∞ (S n , F ). Thus p1 (i + K) ∈ V∞ (S n , F ) × N for any i ∈ N. We define p(i) = p1 (i), ∀i ∈ N + K. Then p is injective, and (2.8) holds. (ii) By Definition 4.2, Lemmas 5.12, 6.1 and Corollary 6.3 of [17], we have ˆi(c1 ) ˆi(c2 ) = 2σ = , (2.10) L(c1 ) L(c2 ) √ √ where σ = lim inf i→∞ i/ 2κi = lim supi→∞ i/ 2κi , L(ck ) denotes the length of the closed geodesic ck , m ˆ k = 1, 2. Note that we have ˆi(cm k ) = mi(ck ) by (2.1) and L(ck ) = mL(ck ). Then (2.10) yields ˆi(cm ˆi(cm 1 ) 2 ) = , m L(c1 ) L(cm 2 )

∀m ∈ N.

k Since κi1 < κi2 by Lemma 2.3, (2.6) and Lemma 2.4, note that L(cm k ) = m1 m2 L(c1 ) < L(c2 ), which, together with (2.11), gives (2.9). 

(2.11) 

k 2E(cm k ) =

√

2κik , then

3. Index iteration theory for closed geodesics on S n In [10] of 1999, Y. 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 [11] of 2000. Note that this index iteration formulae works for Morse indices of iterated closed geodesics (cf. [9], Chap. 12 of [12]). Since every closed geodesic on a sphere must be orientable. Then by Theorem 1.1 of [8] of C. Liu, the initial Morse index of a closed geodesic c on a n-dimensional Finsler sphere coincides with the index of a corresponding symplectic path. As in [10], denote by   λ b N1 (λ, b) = , for λ = ±1, b ∈ R, (3.1) 0 λ   λ 0 D(λ) = , for λ ∈ R \ {0, ±1}, (3.2) 0 λ−1   cos θ − sin θ R(θ) = , for θ ∈ (0, π) ∪ (π, 2π), (3.3) sin θ cos θ   √ R(θ) B , for θ ∈ (0, π) ∪ (π, 2π) and N2 (eθ −1 , B) = 0 R(θ)   b1 b2 B= with bj ∈ R, and b2 ̸= b3 . (3.4) b3 b4 √

Here N2 (eθ −1 , B) is non-trivial if (b2 − b3 ) sin θ < 0, and trivial if (b2 − b3 ) sin θ > 0. As in [10], the -sum (direct sum) of any two real matrices is defined by   A1 0 B 1 0     0 A A1 B1 A2 B2 0 B2    2  = . C1 0 D1 0  C1 D1 C2 D2 2i×2i 2j×2j 0 C2 0 D2 For every M ∈ Sp(2n), the homotopy set Ω (M ) of M in Sp(2n) is defined by Ω (M ) = {N ∈ Sp(2n) | σ(N ) ∩ U = σ(M ) ∩ U ≡ Γ and νω (N ) = νω (M ) ∀ω ∈ Γ },

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where σ(M ) denotes the spectrum of M , νω (M ) ≡ dimC kerC (M − ωI) for ω ∈ U. The component Ω 0 (M ) of P in Sp(2n) is defined by the path connected component of Ω (M ) containing M . Lemma 3.1 (Cf. [10], Lemma 9.1.5 and List 9.1.12 of [12]). For M ∈ Sp(2n) and ω ∈ U, the splitting ± number SM (ω) (cf. Definition 9.1.4 of [12]) satisfies ± SM (ω) = 0,

if ω ∈ ̸ σ(M ).  1, if a ≥ 0, + SN (1) = 1 (1,a) 0, if a < 0.

(3.5) (3.6)

For any Mi ∈ Sp(2ni ) with i = 0 and 1, there holds ± ± ± SM (ω) = SM (ω) + SM (ω), 0 M1 0 1

∀ ω ∈ U.

(3.7)

We have the following Theorem 3.2 (Cf. [11] and Theorem 1.8.10 of [12]). For any M ∈ Sp(2n), there is a path f : [0, 1] → Ω 0 (M ) such that f (0) = M and f (1) = M1  · · ·  Mk ,

(3.8)

where each Mi is a basic normal form listed in (3.1)–(3.4) for 1 ≤ i ≤ k. The following is the common index jump theorem of Y. Long and C. Zhu. Theorem 3.3 (Cf. Theorems 4.1–4.3 of [15]). Let γk , k = 1, . . . , q be a finite collection of symplectic paths. Let Mk = γk (τk ) ∈ Sp(2n). Suppose ˆi(γk , 1) > 0, for all k = 1, . . . , q. Then there exist infinitely many (N, m1 , . . . , mq ) ∈ Nq+1 such that ν(γk , 2mk − 1) = ν(γk , 1),

(3.9)

ν(γk , 2mk + 1) = ν(γk , 1),

(3.10) 

i(γk , 2mk − 1) + ν(γk , 2mk − 1) = 2N − i(γk , 1) +

+ 2SM (1) k

i(γk , 2mk + 1) = 2N + i(γk , 1),



− ν(γk , 1) ,

(3.11) (3.12)

e(Mk ) ≥ 2N − n, 2 e(Mk ) i(γk , 2mk ) + ν(γk , 2mk ) ≤ 2N + ≤ 2N + n, 2 for every k = 1, . . . , q. Moreover we have     mk θ mk θ min , 1− < δ, π π i(γk , 2mk ) ≥ 2N −

(3.13) (3.14)

(3.15)

√

whenever e −1θ ∈ σ(Mk ) and δ can be chosen as small as we want (cf. (4.43) of [15]). More precisely, by (4.10) and (4.40) in [15], we have    N mk = + χk M, 1 ≤ k ≤ q, (3.16) M ˆi(γk , 1) √

where χk = 0 or 1 for 1 ≤ k ≤ q and Mπθ ∈ Z whenever e −1θ ∈ σ(Mk ) and πθ ∈ Q for some 1 ≤ k ≤ q. Furthermore, given M0 ∈ N, by the proof of Theorem 4.1 of [15], we may further require M0 |N (since the closure of the set {{N v} : N ∈ N, M0 |N } is still a closed additive subgroup of Th for some h ∈ N, where we use notations as (4.21) in [15]. Then we can use the proof of Step 2 in Theorem 4.1 of [15] to get N ).

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√

e

−1θj



− SM (e i

√

−1θ

) for 1 ≤ i ≤ q and αi,j =  ∈ σ(Mi ) for 1 ≤ j ≤ µi and 1 ≤ i ≤ q. As in (4.21) of [15], let h = q + 1≤i≤q µi and   αq,µq α1,2 α1,µ1 α2,1 1 α1,1 1 , ,... , ,..., . v= ,..., , ˆi(y1 , 1) ˆi(y2 , 1) ˆi(yq , 1) M ˆi(y1 , 1) M ˆi(yq , 1) ˆi(y1 , 1) ˆi(y1 , 1)

In fact, by (4.40)–(4.41) of [15], let µi =

θ∈(0,2π)

θj π

where

(3.17)

Then by (4.22) of [15], the common index jump theorem is equivalent to find a vertex χ = (χ1 , . . . , χq , χ1,1 , χ1,2 , . . . , χ1,µ1 , χ2,1 , . . . , χq,µq )

(3.18)

of the cube [0, 1]h and infinitely many integers N ∈ N such that |{N v} − χ| < ϵ

(3.19)

for any given ϵ small enough. Theorem 3.4 (Cf. Theorem 4.2 of [15]). Let H be the closure of {{mv}|m ∈ N} in Th = (R/Z)h and V = T0 π −1 H be the tangent space of π −1 H at the origin in Rh , where π : Rh → Th is the projection map. Define A(v) = V \ ∪vk ∈R\Q {x = (x1 , . . . , xh ) ∈ V |xk = 0}.

(3.20)

Define ψ(x) = 0 when x ≥ 0 and ψ(x) = 1 when x < 0. Then for any a = (a1 , . . . , ah ) ∈ A(V ), the vector χ = (ψ(a1 ), . . . , ψ(ah ))

(3.21)

makes (3.19) hold for infinitely many N ∈ N. The following new observation was obtained in [6]. v

Lemma 3.5 (Cf. Lemma 2.9 of [6]). Let v = (v1 , v2 , . . . , vh ) given by (3.17). If vi , vj ∈ R\Q and vji = pq x ∈ Q+ ( i < j), then for ∀x = (x1 , . . . , xh ) ∈ V , we have xji = pq and for ∀a ∈ A(v), we have χi (a) = χj (a). 4. Proof of the main theorems In this section, we give the proofs of Theorems 1.1 and 1.2 by using the techniques similar to those in the proof of Theorem 1.2 in [6]. 2  λ < K ≤ 1, then every nonconstant First note that if the flag curvature K of (S n , F ) satisfies λ+1 closed geodesic c must satisfy i(c) ≥ n − 1,

(4.1)

ˆi(c) > n − 1,

(4.2)

where (4.1) follows from Theorem 3 and Lemma 3 of [18], (4.2) follows from Lemma 2 of [19]. Now it follows from Theorem 2.2 of [15] that e(Pc ) ≥ 0, ∀m ∈ N. (4.3) 2 Here the last inequality holds by (4.1) and the fact that e(Pc ) ≤ 2(n − 1), where Pc ∈ Sp(2n − 2) denotes the linearized Poincar´e map of c. In the rest of this paper, we will assume the following (F) There are only finitely many prime closed geodesics on a Finsler n-sphere (S n , F ) with  2 λ < K ≤ 1. λ+1 i(cm+1 ) − i(cm ) − ν(cm ) ≥ i(c) −

44

H. Duan, H. Liu / Nonlinear Analysis 128 (2015) 36–47

By Theorem 2.7, there exists an integer K ≥ 0 and an injection map p : N + K −→ V∞ (S n , F ) × N such that i(cm ) ≤ 2i + n − 1 ≤ i(cm ) + ν(cm ) holds for any i ∈ N + K, c ∈ CG(S n , F ) and p(i) = (c, m). Denote the elements in V∞ (S n , F ) by V∞ (S n , F ) = {cj | j = 1, 2, . . . , q}. By (4.2) and Theorem 3.3, there exist infinitely many (N, m1 , . . . , mq ) ∈ Nq+1 such that 2mj −1

i(cj

2mj +1

i(cj

θ π



N Mˆi(cj

) = 2N − (i(cj ) + 2SP+c (1) − ν(cj )),

(4.4)

j

) = 2N + i(cj ),

(4.5)

e(Pcj ) ≥ 2N − (n − 1), (4.6) 2 e(Pcj ) 2m 2m i(cj j ) + ν(cj j ) ≤ 2N + ≤ 2N + (n − 1), (4.7) 2   √ Mθ −1θ ∈ σ(Pcj ) and + χ ∈ Z whenever e M , χ = 0 or 1 for 1 ≤ j ≤ q and j j π ) 2mj

i(cj

where mj =

2mj −1

) + ν(cj

) ≥ 2N −

∈ Q for some 1 ≤ j ≤ q. We define λ1 = min (i(cj ) + 2SP+c (1) − ν(cj ) − 1),

(4.8)

λ2 = min i(cj ),

(4.9)

1≤j≤q

j

1≤j≤q

ϱ(S n , F ) =





λ1 + n − 1 . 2

(4.10)

λ2 ≥ n − 1.

(4.11)

By (4.1), we have

Set p(N − s) = (cj(s) , m(s)) with j(s) ∈ {1, . . . , q} and m(s) ∈ N for s = 1, . . . , ϱ(S n , F ). Then by definition of map p in Theorem 2.7, m(s)

m(s)

m(s)

i(cj(s) ) ≤ 2N − 2s + n − 1 ≤ i(cj(s) ) + ν(cj(s) ).

(4.12)

By (4.10) and (4.11), 

 λ1 + n − 1 +n−1 2 ≤ 2N − 2s + n − 1

2N − λ1 ≤ 2N − 2

≤ 2N − 2 + n − 1 ∀ s = 1, . . . , ϱ(S n , F ).

< 2N + λ2 ,

(4.13)

By (4.4)–(4.6), (4.12) and (4.13), we obtain 2m

i(cj(s)j(s)

−1

2m

) + ν(cj(s)j(s)

−1

) < 2N − λ1 ≤ 2N − 2s + n − 1 m(s)

m(s)

≤ i(cj(s) ) + ν(cj(s) ),

(4.14)

and m(s)

2m

i(cj(s) ) ≤ 2N − 2s + n − 1 < 2N + λ2 ≤ i(cj(s)j(s)

+1

).

(4.15)

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H. Duan, H. Liu / Nonlinear Analysis 128 (2015) 36–47

Comparing (4.3), (4.14) and (4.15), we get 2mj(s) − 1 < m(s) < 2mj(s) + 1.

(4.16)

Hence m(s) = 2mj(s) , i.e., for s = 1, . . . , ϱ(S n , F ).

p(N − s) = (cj(s) , 2mj(s) ),

(4.17)

Since the map p is injective, then these j(s)’s are mutually different for s = 1, . . . , ϱ(S n , F ). So q ≥ ϱ(S n , F ).

Lemma 4.1.

ˆi(cj(s ) ) 1 ˆi(cj(s ) ) 2

∈ R\Q for any 1 ≤ s1 < s2 ≤ ϱ(S n , F ).

Proof of Lemma 4.1. Suppose the contrary, i.e.,

ˆi(cj(s ) ) 1 ˆi(cj(s ) ) 2

carry out this proof in two cases as follows:

=

p1 q1

∈ Q. Then

ˆi(cj(s ) ) 1 ˆi(cj(s ) ) 2

∈ Q+ by (4.2). Next we

Case 1: ˆi(cj(s1 ) ) ∈ Q. In this case, we also have ˆi(cj(s2 ) ) ∈ Q. Then by Theorem 3.3, we can require that N ∈ N in (4.4)–(4.7) further satisfies N ∈N M ˆi(cj(sk ) )

and χj(sk ) = 0,

for k = 1, 2.

(4.18)

Then it yields   N + χj(s1 ) M ˆi(cj(s1 ) ) M ˆi(cj(s1 ) )    N 2m 2 ) + χj(s2 ) M ˆi(cj(s2 ) ) = ˆi(cj(s2j(s = 2N = 2 ) ). ˆ M i(cj(s2 ) )

ˆi(c2mj(s1 ) ) = 2 j(s1 )



2m

(4.19)

2m

j(s2 ) 1) ˆ On the other hand, by (2.9) and (4.17), we have ˆi(cj(s1j(s ) ) > i(cj(s2 ) ) which contradicts to (4.19).

Case 2: ˆi(cj(s1 ) ) ∈ R\Q. In this case, we have ˆi(cj(s2 ) ) ∈ R\Q and  v=

ˆi(cj(s ) ) 1 ˆi(cj(s ) ) 2

=

p1 q1

∈ Q. Let

αq,µq 1 1 α1,1 α1,2 α1,µ1 α2,1 ,..., , , ,... , ,..., ˆ ˆ ˆ ˆ ˆ ˆ ˆi(cq ) M i(c1 ) M i(cq ) i(c1 ) i(c1 ) i(c1 ) i(c2 )



given by (3.17), then we have vj(s2 ) = vj(s1 )

1 Mˆi(cj(s2 ) ) 1 Mˆi(cj(s1 ) )

=

ˆi(cj(s ) ) p1 1 = . ˆi(cj(s ) ) q1 2

(4.20)

Note that by Theorem 3.4, for fixed a ∈ A(V ), we can choose N ∈ N such that {N v} − χ(a) is small enough and {N v} − χ(a) ∈ V . Using Lemma 3.5, we obtain   N − χj(s2 )(a) ˆ {N vj(s2 ) } − χj(s2 )(a) p1 M i(cj(s2 ) )  = (4.21) = . N {N vj(s1 ) } − χj(s1 )(a) q1 − χ j(s )(a) 1 Mˆi(c ) j(s1 )

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H. Duan, H. Liu / Nonlinear Analysis 128 (2015) 36–47

Combining (4.20) and (4.21), we get       N N − χj(s1 )(a) M ˆi(cj(s1 ) ) = − χj(s2 )(a) M ˆi(cj(s2 ) ). M ˆi(cj(s1 ) ) M ˆi(cj(s2 ) )

(4.22)

On the other hand, by (2.9) and (4.20), we have    N 2m 1 ) 2 + χj(s1 ) M ˆi(cj(s1 ) ) = ˆi(cj(s1j(s ) ) M ˆi(cj(s1 ) ) 2m

2) > ˆi(cj(s2j(s ) )    N =2 + χj(s2 ) M ˆi(cj(s2 ) ). M ˆi(cj(s2 ) )

Hence, we get 

N M ˆi(cj(s1 ) )



 − χj(s1 )(a)

 M ˆi(cj(s1 ) ) <

which contradicts to (4.22). This proves Lemma 4.1.

N M ˆi(cj(s2 ) )



(4.23)

 − χj(s2 )(a)

M ˆi(cj(s2 ) ),



Now we prove Theorems 1.1 and 1.2. Proof of Theorem 1.1. By Theorem 3.2, Pcj can be connected to pj,0

N1 (1, 1)pj,−  I2

 N1 (1, −1)pj,+  Gj ,

1≤j≤q

(4.24)

in Ω 0 (Pcj ) for some nonnegative integers pj,− , pj,0 , pj,+ , and some symplectic matrix Gj satisfying 1 ̸∈ σ(Gj ). By Lemma 3.1, we have 2SP+c (1) − ν1 (Pcj ) = pj,− − pj,+ ≥ −pj,+ ≥ 1 − n,

1 ≤ j ≤ q.

(4.25)

j

Combining (4.1) with (4.25), we obtain λ1 = min (i(cj ) + 2SP+c (1) − ν(cj ) − 1) ≥ −1, 1≤j≤q

ϱ(S n , F ) ≥

j

which, together with Lemma 4.1, yields the desired result.

n 2

− 1,



Proof of Theorem 1.2. Under the bumpy condition, we have pj,− = pj,0 = pj,+ = 0 in (4.24) for 1 ≤ j ≤ q. Then by Lemma 3.1, we have 2SP+c (1) − ν1 (Pcj ) = pj,− − pj,+ ≥ −pj,+ ≥ 0,

1 ≤ j ≤ q.

(4.26)

j

Combining (4.1) with (4.26), we obtain λ1 = min (i(cj ) + 2SP+c (1) − ν(cj ) − 1) ≥ n − 2, 1≤j≤q

ϱ(S n , F ) ≥ n − 2,

j

which, together with Lemma 4.1, yields the desired result.



Acknowledgments The authors sincerely thank the referees for their valuable comments and suggestions on this paper. First author was partially supported by NSFC (No. 11131004, 11471169). Second author was partially supported by NSFC (No. 11401555), China Postdoctoral Science Foundation No. 2014T70589, CUSF (No. WK3470000001).

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