2 Critical Point Theory for Closed Geodesics
In this section, we describe briefly the critical point theory for closed geodesics.
On a compact Finsler manifold (M,F), we choose an auxiliary Riemannian metric. This endows the space Λ=ΛM of H1-maps γ:S1→M with a natural Riemannian Hilbert manifold structure on which the group S1=ℝ/ℤ acts continuously by isometries, cf. [13, Chapters 1 and 2]. This action is defined by translating the parameter, i.e.,
(
s
⋅
γ
)
(
t
)
=
γ
(
t
+
s
)
for all γ∈Λ and s,t∈S1. The Finsler metric F defines an energy functional E and a length functional L on Λ by
E
(
γ
)
=
1
2
∫
S
1
F
(
γ
˙
(
t
)
)
2
𝑑
t
,
L
(
γ
)
=
∫
S
1
F
(
γ
˙
(
t
)
)
𝑑
t
.
Both functionals are invariant under the S1-action. By [28], the functional E is C1,1 on Λ and satisfies the Palais–Smale condition. Thus we can apply the deformation theorems in [3] and [27]. The critical points of E of positive energies are precisely the closed geodesics c:S1→M of the Finsler structure. If c∈Λ is a closed geodesic, then c is a regular curve, i.e., c˙(t)≠0 for all t∈S1, and this implies that the second differential E′′(c) of E at c exists. As usual we define the index i(c) of c as the maximal dimension of subspaces of TcΛ on which E′′(c) is negative definite, and the nullity ν(c) of c so that ν(c)+1 is the dimension of the null space of E′′(c).
For m∈ℕ we denote the m-fold iteration map ϕm:Λ→Λ by
ϕ
m
(
γ
)
(
t
)
=
γ
(
m
t
)
for all
γ
∈
Λ
,
t
∈
S
1
.
We also use the notation ϕm(γ)=γm. For a closed geodesic c, the average index is defined by
i
^
(
c
)
=
lim
m
→
∞
i
(
c
m
)
m
.
If γ∈Λ is not constant, then the multiplicity m(γ) of γ is the order of the isotropy group {s∈S1:s⋅γ=γ}. If m(γ)=1, then γ is called prime. Hence m(γ)=m if and only if there exists a prime curve γ~∈Λ such that γ=γ~m.
In this paper for κ∈ℝ we denote
Λ
κ
=
{
d
∈
Λ
:
E
(
d
)
≤
κ
}
.
For a closed geodesic c we set
Λ
(
c
)
=
{
γ
∈
Λ
:
E
(
γ
)
<
E
(
c
)
}
.
We call a closed geodesic satisfying the isolation condition if the following holds:
Condition (Iso).
For all m∈ℕ the orbit S1⋅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).
Using singular homology with rational coefficients, we consider the following critical ℚ-module of a closed geodesic c∈Λ:
C
¯
*
(
E
,
c
)
=
H
*
(
(
Λ
(
c
)
∪
S
1
⋅
c
)
/
S
1
,
Λ
(
c
)
/
S
1
)
.
Proposition 2.1 (cf. [30, Satz 6.11] or [1, Proposition 3.12]).
Let c be a prime closed geodesic on a Finsler manifold (M,F) satisfying (Iso). Then we have
C
¯
q
(
E
,
c
m
)
≡
H
q
(
(
Λ
(
c
m
)
∪
S
1
⋅
c
m
)
/
S
1
,
Λ
(
c
m
)
/
S
1
)
=
(
H
i
(
c
m
)
(
U
c
m
-
∪
{
c
m
}
,
U
c
m
-
)
⊗
H
q
-
i
(
c
m
)
(
N
c
m
-
∪
{
c
m
}
,
N
c
m
-
)
)
+
ℤ
m
.
When
ν
(
c
m
)
=
0
, there holds
C
¯
q
(
E
,
c
m
)
=
{
ℚ
if
i
(
c
m
)
-
i
(
c
)
∈
2
ℤ
and
q
=
i
(
c
m
)
,
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
.
When
ν
(
c
m
)
>
0
, there holds
C
¯
q
(
E
,
c
m
)
=
H
q
-
i
(
c
m
)
(
N
c
m
-
∪
{
c
m
}
,
N
c
m
-
)
(
-
1
)
i
(
c
m
)
-
i
(
c
)
ℤ
m
,
where
N
c
m
is a local characteristic manifold at
c
m
and
N
c
m
-
=
N
c
m
∩
Λ
(
c
m
)
, Ucm is a local negative disk at cm and Ucm-=Ucm∩Λ(cm), H∗(X,A)±ℤm={[ξ]∈H∗(X,A):T∗[ξ]=±[ξ]}, where T is a generator of the ℤm-action.
Denote
k
j
(
c
m
)
≡
dim
H
j
(
N
c
m
-
∪
{
c
m
}
,
N
c
m
-
)
(
-
1
)
i
(
c
m
)
-
i
(
c
)
ℤ
m
.
Clearly the integers kj(cm) are equal to 0 when j<0 or j>ν(cm) and can take only values 0 or 1 when j=0 or j=ν(cm).
Proposition 2.2 (cf. [30, Satz 6.13]).
Let c be a prime closed geodesic on a Finsler manifold (M,F) satisfying (Iso). For any m∈N, we have:
If
k
0
(
c
m
)
=
1
, there holds
k
j
(
c
m
)
=
0
for
1
≤
j
≤
ν
(
c
m
)
.
If
k
ν
(
c
m
)
(
c
m
)
=
1
, there holds
k
j
(
c
m
)
=
0
for
0
≤
j
≤
ν
(
c
m
)
-
1
.
If
k
j
(
c
m
)
≥
1
for some
1
≤
j
≤
ν
(
c
m
)
-
1
, there holds
k
ν
(
c
m
)
(
c
m
)
=
0
=
k
0
(
c
m
)
.
In particular, if
ν
(
c
m
)
≤
2
, then only one of the integers
k
j
(
c
m
)
can be non-zero.
By [37, Lemma 5.2], we have the following periodic property for kl(cm):
Proposition 2.3.
Let c be a prime closed geodesic on a compact Finsler manifold (M,F) satisfying (Iso). Then there exists a minimal T(c)∈N such that
ν
(
c
p
+
T
(
c
)
)
=
ν
(
c
p
)
,
i
(
c
p
+
T
(
c
)
)
-
i
(
c
p
)
∈
2
ℤ
for all
p
∈
ℕ
,
k
l
(
c
p
+
T
(
c
)
)
=
k
l
(
c
p
)
for all
p
∈
ℕ
,
l
∈
ℤ
.
Definition 2.4.
The Euler characteristicχ(cm) of cm is defined by
χ
(
c
m
)
≡
χ
(
(
Λ
(
c
m
)
∪
S
1
⋅
c
m
)
/
S
1
,
Λ
(
c
m
)
/
S
1
)
,
(2.1)
≡
∑
q
=
0
∞
(
-
1
)
q
dim
C
¯
q
(
E
,
c
m
)
=
∑
l
=
0
2
n
-
2
(
-
1
)
i
(
c
m
)
+
l
k
l
(
c
m
)
.
Here χ(A,B) denotes the usual Euler characteristic of the space pair (A,B).
The average Euler characteristicχ^(c) of c is defined by
(2.2)
χ
^
(
c
)
=
lim
N
→
∞
1
N
∑
1
≤
m
≤
N
χ
(
c
m
)
.
By of [37, Remark 5.4], χ^(c) is well-defined and is a rational number, in fact
(2.3)
χ
^
(
c
)
=
1
T
(
c
)
∑
1
≤
m
≤
T
(
c
)
χ
(
c
m
)
.
In particular, if cm are non-degenerate for for all m∈ℕ, then
χ
^
(
c
)
=
{
(
-
1
)
i
(
c
)
if
i
(
c
2
)
-
i
(
c
)
∈
2
ℤ
,
(
-
1
)
i
(
c
)
2
otherwise.
We have the following mean index identity for closed geodesics:
Theorem 2.5 (cf. [30, Theorem 7.9] or [37, Theorem 5.5]).
Suppose that there exist only finitely many prime closed geodesics {cj}1≤j≤p with i^(cj)>0 for 1≤j≤p on (Sn,F). Then the following identity holds:
(2.4)
∑
1
≤
j
≤
p
χ
^
(
c
j
)
i
^
(
c
j
)
=
B
(
n
,
1
)
=
{
-
n
2
n
-
2
,
n
even
,
n
+
1
2
n
-
2
n
odd
.
Set Λ¯0=Λ¯0M={constant point curves in M}≅M. Let (X,Y) be a space pair such that the Betti numbers bi=bi(X,Y)=dimHi(X,Y;ℚ) are finite for all i∈ℤ. As usual the Poincaré series of (X,Y) is defined by the formal power series P(X,Y)=∑i=0∞biti. We need the following results on Betti numbers.
Theorem 2.6 (H.-B. Rademacher [29, Theorem 2.4 and Remark 2.5]).
We have the following Poincaré series:
When
n
=
2
k
+
1
is odd,
P
(
Λ
¯
S
n
,
Λ
¯
0
S
n
)
(
t
)
=
t
n
-
1
(
1
1
-
t
2
+
t
n
-
1
1
-
t
n
-
1
)
=
t
2
k
(
1
1
-
t
2
+
t
2
k
1
-
t
2
k
)
.
Thus for
q
∈
ℤ
and
l
∈
ℕ
0
, we have
b
q
=
b
q
(
Λ
¯
S
n
,
Λ
¯
0
S
n
)
=
rank
H
q
(
Λ
¯
S
n
,
Λ
¯
0
S
n
)
(2.5)
=
{
2
if
q
∈
{
4
k
+
2
l
,
l
=
0
mod
k
}
,
1
if
q
∈
{
2
k
}
∪
{
2
k
+
2
l
,
l
≠
0
mod
k
}
,
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
.
When
n
=
2
k
is even,
P
(
Λ
¯
S
n
,
Λ
¯
0
S
n
)
(
t
)
=
t
n
-
1
(
1
1
-
t
2
+
t
n
(
m
+
1
)
-
2
1
-
t
n
(
m
+
1
)
-
2
)
1
-
t
n
m
1
-
t
n
=
t
2
k
-
1
(
1
1
-
t
2
+
t
4
k
-
2
1
-
t
4
k
-
2
)
,
where
m
=
1
by
[
29
, Theorem 2.4]
. Thus for
q
∈
ℤ
and
l
∈
ℕ
0
, we have
b
q
=
b
q
(
Λ
¯
S
n
,
Λ
¯
0
S
n
)
=
rank
H
q
(
Λ
¯
S
n
,
Λ
¯
0
S
n
)
(2.6)
=
{
2
if
q
∈
{
6
k
-
3
+
2
l
,
l
=
0
mod
2
k
-
1
}
,
1
if
q
∈
{
2
k
-
1
}
∪
{
2
k
-
1
+
2
l
,
l
≠
0
mod
2
k
-
1
}
,
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
.
We have the following version of the Morse inequality.
Theorem 2.7 ([30, Theorem 6.1]).
Suppose that there exist only finitely many prime closed geodesics {cj}1≤j≤p on (M,F), and 0≤a<b≤∞. Define for each q∈Z,
M
q
(
Λ
¯
b
,
Λ
¯
a
)
=
∑
1
≤
j
≤
p
,
a
<
E
(
c
j
m
)
<
b
rank
C
¯
q
(
E
,
c
j
m
)
b
q
(
Λ
¯
b
,
Λ
¯
a
)
=
rank
H
q
(
Λ
¯
b
,
Λ
¯
a
)
.
Then there holds
M
q
(
Λ
¯
b
,
Λ
¯
a
)
-
M
q
-
1
(
Λ
¯
b
,
Λ
¯
a
)
+
⋯
+
(
-
1
)
q
M
0
(
Λ
¯
b
,
Λ
¯
a
)
≥
b
q
(
Λ
¯
b
,
Λ
¯
a
)
-
b
q
-
1
(
Λ
¯
b
,
Λ
¯
a
)
+
⋯
+
(
-
1
)
q
b
0
(
Λ
¯
b
,
Λ
¯
a
)
,
M
q
(
Λ
¯
b
,
Λ
¯
a
)
≥
b
q
(
Λ
¯
b
,
Λ
¯
a
)
.
Next we recall the Fadell–Rabinowitz index in a relative version due to [31]. Let X be an S1-space, A⊂X a closed S1-invariant subset. Note that the cup product defines a homomorphism
H
S
1
∗
(
X
)
⊗
H
S
1
∗
(
X
,
A
)
→
H
S
1
∗
(
X
,
A
)
,
(
ζ
,
z
)
↦
ζ
∪
z
,
where HS1∗ is the S1-equivariant cohomology with rational coefficients in the sense of A. Borel (cf. [2, Chapter IV]). We fix a characteristic class η∈H2(ℂℙ∞). Let f∗:H∗(ℂℙ∞)→HS1∗(X) be the homomorphism induced by a classifying map f:XS1→ℂℙ∞. Now for γ∈H∗(ℂℙ∞) and z∈HS1∗(X,A), let γ⋅z=f∗(γ)∪z. Then the order ordη(z) with respect to η is defined by
ord
η
(
z
)
=
inf
{
k
∈
ℕ
∪
{
∞
}
:
η
k
⋅
z
=
0
}
.
By [31, Proposition 3.1], there is an element z∈HS1n+1(Λ,Λ0) of infinite order, i.e., ordη(z)=∞. For κ≥0, we denote by jκ:(Λκ,Λ0)→(Λ,Λ0) the natural inclusion and define the function dz:ℝ≥0→ℕ∪{∞}:
d
z
(
κ
)
=
ord
η
(
j
κ
∗
(
z
)
)
.
Denote dz(κ-)=limϵ↘0dz(κ-ϵ), where t↘a means t>a and t→a.
Then we have the following property due to [31, Section 5]:
Lemma 2.8 (H.-B. Rademacher).
The function dz is non-decreasing and limλ↘κdz(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, we define
κ
i
=
inf
{
δ
∈
ℝ
:
d
z
(
δ
)
≥
i
}
.
Then we have the following:
Lemma 2.9 (cf. [39, Lemma 2.3]).
Suppose that there are only finitely many prime closed geodesics on (Sn,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 (Sn,F).
Lemma 2.10 (cf. [39, Lemma 2.4]).
Suppose that there are only finitely many prime closed geodesics on (Sn,F). Then for every i∈N, there exists a closed geodesic c on (Sn,F) such that
(2.7)
E
(
c
)
=
κ
i
,
C
¯
2
i
+
dim
(
z
)
-
2
(
E
,
c
)
≠
0
.
Definition 2.11.
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 cinfinitely variationally visible if there exist infinitely many m,i∈ℕ such that c is (m,i)-variationally visible. We denote by 𝒱∞(Sn,F) the set of infinitely variationally visible closed geodesics.
Theorem 2.12 (cf. [39, Theorem 2.6]).
Suppose that there are only finitely many prime closed geodesics on (Sn,F). Then for any c∈V∞(Sn,F), we have
i
^
(
c
)
L
(
c
)
=
2
σ
,
where σ=liminfi→∞i2κi=limsupi→∞i2κi.
3 Index Iteration Theory for Closed Geodesics
In this section, we recall briefly the index theory for symplectic paths developed by Y. Long and his coworkers. All the details can be found in [22]. Then we use this theory to study the Morse indices of closed geodesics.
Let c be a closed geodesic on an orientable Finsler manifold M=(M,F). Denote the linearized Poincaré map of c by Pc∈Sp(2n-2). Then Pc is a symplectic matrix. Note that the index iteration formulae in [21] of 2000 (cf. [22, Chapter 8]) work for Morse indices of iterated closed geodesics (cf. [16], [22, Chapter 12]). Since every closed geodesic on M is orientable. Then by [17, Theorem 1.1] of C. Liu (cf. also [40]), the initial Morse index of a closed geodesic c on M coincides with the index of a corresponding symplectic path introduced by C. Conley, E. Zehnder, and Y. Long in 1984–1990 (cf. [22]).
As usual, the symplectic group Sp(2n) is defined by
Sp
(
2
n
)
=
{
M
∈
GL
(
2
n
,
ℝ
)
:
M
T
J
M
=
J
}
,
whose topology is induced from that of ℝ4n2. For τ>0 we are interested in paths in Sp(2n):
𝒫
τ
(
2
n
)
=
{
γ
∈
C
(
[
0
,
τ
]
,
Sp
(
2
n
)
)
:
γ
(
0
)
=
I
2
n
}
,
which is equipped with the topology induced from that of Sp(2n). The following real function was introduced in [21]:
D
ω
(
M
)
=
(
-
1
)
n
-
1
ω
¯
n
det
(
M
-
ω
I
2
n
)
for all
ω
∈
𝕌
,
M
∈
Sp
(
2
n
)
.
Thus for any ω∈𝕌 the following codimension-1 hypersurface in Sp(2n) is defined in [21]:
Sp
(
2
n
)
ω
0
=
{
M
∈
Sp
(
2
n
)
:
D
ω
(
M
)
=
0
}
.
For any M∈Sp(2n)ω0, we define a co-orientation of Sp(2n)ω0 at M by the positive direction ddtMetϵJ|t=0 of the path MetϵJ with 0≤t≤1 and ϵ>0 being sufficiently small. Let
Sp
(
2
n
)
ω
∗
=
Sp
(
2
n
)
∖
Sp
(
2
n
)
ω
0
,
𝒫
τ
,
ω
∗
(
2
n
)
=
{
γ
∈
𝒫
τ
(
2
n
)
:
γ
(
τ
)
∈
Sp
(
2
n
)
ω
∗
}
,
𝒫
τ
,
ω
0
(
2
n
)
=
𝒫
τ
(
2
n
)
∖
𝒫
τ
,
ω
∗
(
2
n
)
.
For any two continuous arcs ξ and η:[0,τ]→Sp(2n) with ξ(τ)=η(0), it is defined as usual:
η
∗
ξ
(
t
)
=
{
ξ
(
2
t
)
if
0
≤
t
≤
τ
2
,
η
(
2
t
-
τ
)
if
τ
2
≤
t
≤
τ
.
Given any two 2mk×2mk matrices of square block form
M
k
=
(
A
k
B
k
C
k
D
k
)
with k=1,2, as in [22], the ⋄-product of M1 and M2 is defined by the following 2(m1+m2)×2(m1+m2) matrix M1⋄M2:
M
1
⋄
M
2
=
(
A
1
0
B
1
0
0
A
2
0
B
2
C
1
0
D
1
0
0
C
2
0
D
2
)
.
Denote by M⋄k the k-fold ⋄-product M⋄⋯⋄M. Note that the ⋄-product of any two symplectic matrices is symplectic. For any two paths γj∈𝒫τ(2nj) with j=0 and 1, let
γ
0
⋄
γ
1
(
t
)
=
γ
0
(
t
)
⋄
γ
1
(
t
)
for all t∈[0,τ].
A special path ξn is defined by
ξ
n
(
t
)
=
(
2
-
t
τ
0
0
(
2
-
t
τ
)
-
1
)
⋄
n
for
0
≤
t
≤
τ
.
Definition 3.1 (cf. [21, 22]).
For any ω∈𝕌 and M∈Sp(2n), define
ν
ω
(
M
)
=
dim
ℂ
ker
ℂ
(
M
-
ω
I
2
n
)
.
For any τ>0 and γ∈𝒫τ(2n), define
ν
ω
(
γ
)
=
ν
ω
(
γ
(
τ
)
)
.
If γ∈𝒫τ,ω∗(2n), define
(3.1)
i
ω
(
γ
)
=
[
Sp
(
2
n
)
ω
0
:
γ
∗
ξ
n
]
,
where the right-hand side of (3.1) is the usual homotopy intersection number, and the orientation of γ∗ξn is its positive time direction under homotopy with fixed end points.
If γ∈𝒫τ,ω0(2n), we let ℱ(γ) be the set of all open neighborhoods of γ in 𝒫τ(2n), and define
i
ω
(
γ
)
=
sup
U
∈
ℱ
(
γ
)
inf
{
i
ω
(
β
)
:
β
∈
U
∩
𝒫
τ
,
ω
∗
(
2
n
)
}
.
Then
(
i
ω
(
γ
)
,
ν
ω
(
γ
)
)
∈
ℤ
×
{
0
,
1
,
…
,
2
n
}
is called the index function of γ at ω.
For any symplectic path γ∈𝒫τ(2n) and m∈ℕ, we define its m-th iteration γm:[0,mτ]→Sp(2n) by
γ
m
(
t
)
=
γ
(
t
-
j
τ
)
γ
(
τ
)
j
for
j
τ
≤
t
≤
(
j
+
1
)
τ
,
j
=
0
,
1
,
…
,
m
-
1
.
We still denote the extended path on [0,+∞) by γ.
Definition 3.2 (cf. [21, 22]).
For any γ∈𝒫τ(2n), we define
(
i
(
γ
,
m
)
,
ν
(
γ
,
m
)
)
=
(
i
1
(
γ
m
)
,
ν
1
(
γ
m
)
)
for all
m
∈
ℕ
.
The mean indexi^(γ,m) per mτ for m∈ℕ is defined by
i
^
(
γ
,
m
)
=
lim
k
→
+
∞
i
(
γ
,
m
k
)
k
.
For any M∈Sp(2n) and ω∈𝕌, the splitting numbersSM±(ω) of M at ω are defined by
(3.2)
S
M
±
(
ω
)
=
lim
ϵ
→
0
+
i
ω
exp
(
±
-
1
ϵ
)
(
γ
)
-
i
ω
(
γ
)
for any path γ∈𝒫τ(2n) satisfying γ(τ)=M.
For a given path γ∈𝒫τ(2n) we consider to deform it to a new path η in 𝒫τ(2n) so that
(3.3)
i
1
(
γ
m
)
=
i
1
(
η
m
)
,
ν
1
(
γ
m
)
=
ν
1
(
η
m
)
for all
m
∈
ℕ
,
and that (i1(ηm),ν1(ηm)) is easy enough to compute. This leads to finding homotopies
δ
:
[
0
,
1
]
×
[
0
,
τ
]
→
Sp
(
2
n
)
starting from γ in 𝒫τ(2n) and keeping the end points of the homotopy always stay in a certain suitably chosen maximal subset of Sp(2n) so that (3.3) always holds. In fact, this set was first discovered in [21] as the path connected component Ω0(M) containing M=γ(τ) of the set
Ω
(
M
)
=
{
N
∈
Sp
(
2
n
)
:
σ
(
N
)
∩
𝕌
=
σ
(
M
)
∩
𝕌
and
ν
λ
(
N
)
=
ν
λ
(
M
)
for all
λ
∈
σ
(
M
)
∩
𝕌
}
.
Here Ω0(M) is called the homotopy component of M in Sp(2n).
In [21] and [22], the following symplectic matrices were introduced as basic normal forms:
D
(
λ
)
=
(
λ
0
0
λ
-
1
)
,
λ
=
±
2
,
N
1
(
λ
,
b
)
=
(
λ
b
0
λ
)
,
λ
=
±
1
,
b
=
±
1
,
0
,
R
(
θ
)
=
(
cos
θ
-
sin
θ
sin
θ
cos
θ
)
,
θ
∈
(
0
,
π
)
∪
(
π
,
2
π
)
,
N
2
(
ω
,
b
)
=
(
R
(
θ
)
b
0
R
(
θ
)
)
,
θ
∈
(
0
,
π
)
∪
(
π
,
2
π
)
,
where b=(b1b2b3b4) with bi∈ℝ and b2≠b3. We call N2(ω,b)trivial if (b2-b3)sinθ>0, N2(ω,b)non-trivial if (b2-b3)sinθ<0.
Splitting numbers possess the following properties:
Lemma 3.3 (cf. [21] and [22, Lemma 9.1.5]).
Splitting numbers SM±(ω) are well defined, i.e., they are independent of the choice of the path γ∈Pτ(2n) satisfying γ(τ)=M appeared in (3.2). For ω∈U and M∈Sp(2n), splitting numbers SN±(ω) are constant for all N∈Ω0(M).
Lemma 3.4 (cf. [21], [22, Lemma 9.1.5 and List 9.1.12]).
For M∈Sp(2n) and ω∈U, there holds
S
M
±
(
ω
)
=
0
if
ω
∉
σ
(
M
)
,
S
N
1
(
1
,
a
)
+
(
1
)
=
{
1
if
a
≥
0
,
0
if
a
<
0
.
For any Mi∈Sp(2ni) with i=0 and 1, there holds
S
M
0
⋄
M
1
±
(
ω
)
=
S
M
0
±
(
ω
)
+
S
M
1
±
(
ω
)
for all
ω
∈
𝕌
.
The following theorem contains the precise index iteration formulae for symplectic paths, which is due to Y. Long (cf. [22, Theorem 8.3.1 and Corollary 8.3.2]).
Theorem 3.5.
Let γ∈Pτ(2n). Then there exists a path f∈C([0,1],Ω0(γ(τ)) such that f(0)=γ(τ) and
f
(
1
)
=
N
1
(
1
,
1
)
⋄
p
-
⋄
I
2
p
0
⋄
N
1
(
1
,
-
1
)
⋄
p
+
⋄
N
1
(
-
1
,
1
)
⋄
q
-
⋄
(
-
I
2
q
0
)
⋄
N
1
(
-
1
,
-
1
)
⋄
q
+
⋄
R
(
θ
1
)
⋄
⋯
⋄
R
(
θ
r
)
⋄
N
2
(
ω
1
,
u
1
)
⋄
⋯
⋄
N
2
(
ω
r
*
,
u
r
*
)
(3.4)
⋄
N
2
(
λ
1
,
v
1
)
⋄
⋯
⋄
N
2
(
λ
r
0
,
v
r
0
)
⋄
M
0
,
where the N2(ωj,uj) are non-trivial and the N2(λj,vj) are trivial basic normal forms; σ(M0)∩U=∅; p-, p0, p+, q-, q0, q+, r, r* and r0 are non-negative integers; ωj=e-1αj, λj=e-1βj; θj, αj, βj∈(0,π)∪(π,2π); these integers and real numbers are uniquely determined by γ(τ). Then using the functions defined in (1.1),
i
(
γ
,
m
)
=
m
(
i
(
γ
,
1
)
+
p
-
+
p
0
-
r
)
+
2
∑
j
=
1
r
E
(
m
θ
j
2
π
)
-
r
-
p
-
-
p
0
(3.5)
-
1
+
(
-
1
)
m
2
(
q
0
+
q
+
)
+
2
(
∑
j
=
1
r
*
φ
(
m
α
j
2
π
)
-
r
*
)
,
ν
(
γ
,
m
)
=
ν
(
γ
,
1
)
+
1
+
(
-
1
)
m
2
(
q
-
+
2
q
0
+
q
+
)
+
2
(
r
+
r
*
+
r
0
)
(3.6)
-
2
(
∑
j
=
1
r
φ
(
m
θ
j
2
π
)
+
∑
j
=
1
r
*
φ
(
m
α
j
2
π
)
+
∑
j
=
1
r
0
φ
(
m
β
j
2
π
)
)
,
(3.7)
i
^
(
γ
,
1
)
=
i
(
γ
,
1
)
+
p
-
+
p
0
-
r
+
∑
j
=
1
r
θ
j
π
.
We have i(γ,1) is odd if f(1)=N1(1,1), I2, N1(-1,1), -I2, N1(-1,-1) and R(θ); i(γ,1) is even if f(1)=N1(1,-1) and N2(ω,b); i(γ,1) can be any integer if σ(f(1))∩U=∅.
We have the following properties in the index iteration theory.
Theorem 3.6 (cf. [25, Theorem 2.2]).
Let γ∈Pτ(2n) and M=γ(τ). Then for any m∈N, there holds
ν
(
γ
,
m
)
-
e
(
M
)
2
≤
i
(
γ
,
m
+
1
)
-
i
(
γ
,
m
)
-
i
(
γ
,
1
)
≤
ν
(
γ
,
1
)
-
ν
(
γ
,
m
+
1
)
+
e
(
M
)
2
,
where e(M) is the total algebraic multiplicity of all eigenvalues of M on the unit circle in the complex plane C.
The following is the common index jump theorem of Y. Long and C. Zhu.
Theorem 3.7 (cf. [25, Theorems 4.1–4.3]).
Let γk∈Pτk(2n), where k=1,…,q, be a finite collection of symplectic paths. Let Mk=γk(τk). Suppose that i^(γk,1)>0 for all k=1,…,q. Then there exist infinitely many (T,m1,…,mq)∈Nq+1 such that
(3.8)
ν
(
γ
k
,
2
m
k
-
1
)
=
ν
(
γ
k
,
1
)
,
(3.9)
ν
(
γ
k
,
2
m
k
+
1
)
=
ν
(
γ
k
,
1
)
,
(3.10)
i
(
γ
k
,
2
m
k
-
1
)
+
ν
(
γ
k
,
2
m
k
-
1
)
=
2
T
-
(
i
(
γ
k
,
1
)
+
2
S
M
k
+
(
1
)
-
ν
(
γ
k
,
1
)
)
,
(3.11)
i
(
γ
k
,
2
m
k
+
1
)
=
2
T
+
i
(
γ
k
,
1
)
,
(3.12)
i
(
γ
k
,
2
m
k
)
≥
2
T
-
e
(
M
k
)
2
≥
2
T
-
n
,
(3.13)
i
(
γ
k
,
2
m
k
)
+
ν
(
γ
k
,
2
m
k
)
≤
2
T
+
e
(
M
k
)
2
≤
2
T
+
n
for every k=1,…,q. Moreover, we have
(3.14)
min
{
{
m
k
θ
π
}
,
1
-
{
m
k
θ
π
}
}
<
δ
whenever e-1θ∈σ(Mk) and δ can be chosen as small as we want (cf. [25, (4.43)]).
More precisely, by [25, (4.10) and (4.40) ], we have
m
k
=
(
[
T
M
i
^
(
γ
k
,
1
)
]
+
χ
k
)
M
,
1
≤
k
≤
q
,
where χk=0 or 1 for 1≤k≤q and Mθπ∈ℤ whenever e-1θ∈σ(Mk) and θπ∈ℚ for some 1≤k≤q. Furthermore, given M0∈ℕ, by [25, proof of Theorem 4.1], we may further require M0∣T (since the closure of the set {{Tv}:T∈ℕ,M0|T} is still a closed additive subgroup of 𝕋h for some h∈ℕ, where we use notations as [25, (4.21)]. Then we can use [25, proof of Step 2 in Theorem 4.1] to get T). In fact, let
μ
i
=
∑
θ
∈
(
0
,
2
π
)
S
M
i
-
(
e
-
1
θ
)
for
1
≤
i
≤
q
and αi,j=θjπ, where e-1θj∈σ(Mi) for 1≤j≤μi and 1≤i≤q. Let
h
=
q
+
∑
1
≤
i
≤
q
μ
i
and
(3.15)
v
=
(
1
M
i
^
(
γ
1
,
1
)
,
…
,
1
M
i
^
(
γ
q
,
1
)
,
α
1
,
1
i
^
(
γ
1
,
1
)
,
α
1
,
2
i
^
(
γ
1
,
1
)
,
…
,
α
1
,
μ
1
i
^
(
γ
1
,
1
)
,
α
2
,
1
i
^
(
γ
2
,
1
)
,
…
,
α
q
,
μ
q
i
^
(
γ
q
,
1
)
)
∈
ℝ
h
.
Then the above theorem is equivalent to find a vertex
χ
=
(
χ
1
,
…
,
χ
q
,
χ
1
,
1
,
χ
1
,
2
,
…
,
χ
1
,
μ
1
,
χ
2
,
1
,
…
,
χ
q
,
μ
q
)
of the cube [0,1]h and infinitely many integers T∈ℕ such that
(3.16)
|
{
T
v
}
-
χ
|
<
ϵ
for any given ϵ small enough (cf. [25, p. 346 and p. 349]).
Theorem 3.8 (cf. [25, Theorem 4.2]).
Let H be the closure of the subset {{mv}:m∈N} in Th=(R/Z)h and let V=T0π-1H be the tangent space of π-1H at the origin in Rh, where π:Rh→Th is the projection map. Define
A
(
v
)
=
V
∖
⋃
v
k
∈
ℝ
∖
ℚ
{
x
=
(
x
1
,
,
…
,
x
h
)
∈
V
:
x
k
=
0
}
.
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
(
a
1
)
,
…
,
ψ
(
a
h
)
)
makes (3.16) hold for infinitely many T∈N.
Theorem 3.9 (cf. [25, Theorem 4.2]).
We have the following properties for A(v):
When
v
∈
ℝ
h
∖
ℚ
h
, then
dim
V
≥
1
, 0∉A(v)⊂V, A(v)=-A(v) and A(v) is open in V.
When
dim
V
=
1
, then
A
(
v
)
=
V
∖
{
0
}
.
When
dim
V
≥
2
, A(v) is obtained from V by deleting all the coordinate hyperplanes with dimension strictly smaller than dimV from V.
4 Proof of the Main Theorems
In this section, we give the proofs of the main theorems. In the rest of this paper, we assume the following:
Assumption (F).
There are only finitely many prime closed geodesics {cj}1≤j≤p on (M,F).
Let π:M~→M be the universal covering space of M and F~=π∗F. Then (M~,F~) is a Finsler manifold that is locally isometric to (M,F). Thus the flag curvature K of (M~,F~) satisfies (λλ+1)2<K≤1 by assumption. Hence by [32], M~ is homeomorphic to Sn.
For each prime closed geodesic cj on (M,F), clearly there exists a minimal αj∈ℕ such that cjαj lifts to a prime closed geodesic on (M~,F~). Denote by {c~j,1,…,c~j,nj} the lifts of cjαj for some nj∈ℕ such that the c~jl are pairwise distinct for 1≤l≤nj. For each l∈{2,…,nj}, there is a covering transformation h:M~→M~ such that h(c~j,l)=c~j,1. By the definition of F~, h is an isometry on (M~,F~). Therefore h preserves the energy functional, i.e., E(γ)=E(h(γ)) for any γ∈ΛM~. In particular, we have
(4.1)
i
(
c
~
j
,
l
m
)
=
i
(
c
~
j
,
1
m
)
,
ν
(
c
~
j
,
l
m
)
=
ν
(
c
~
j
,
1
m
)
,
C
¯
q
(
E
,
c
~
j
,
l
m
)
≅
C
¯
q
(
E
,
c
~
j
,
1
m
)
for all
m
∈
ℕ
,
q
∈
ℤ
,
2
≤
l
≤
n
j
.
Hence there are exactly q=∑1≤j≤pnj prime closed geodesics {c~1,1,…,c~1,n1,…,c~p,1,…,c~p,np} on (M~,F~). Denote by {Pc~j,l}1≤j≤p, 1≤l≤nj the linearized Poincaré maps of {c~j,l}1≤j≤p, 1≤l≤nj. Then
P
c
~
j
,
l
=
P
c
~
j
,
1
for 1≤j≤p and 2≤l≤nj.
Since the flag curvature K of (M~,F~) satisfies (λλ+1)2<K≤1 by assumption, every non-constant closed geodesic c~ on (M~,F~) must satisfy
(4.2)
i
(
c
~
)
≥
n
-
1
by [32, Theorem 3 and Lemma 3]. Now it follows from Theorem 3.6 and (4.2) that
(4.3)
i
(
c
~
j
,
l
m
+
1
)
-
i
(
c
~
j
,
l
m
)
-
ν
(
c
~
j
,
l
m
)
≥
i
(
c
~
j
,
l
)
-
e
(
P
c
~
j
,
l
)
2
≥
0
,
1
≤
j
≤
p
,
1
≤
l
≤
n
j
,
for all
m
∈
ℕ
.
Here the last inequality holds by (4.2) and the fact that e(Pc~j,l)≤2(n-1).
Note that we have
i
^
(
c
~
j
,
l
)
>
n
-
1
for 1≤j≤p, 1≤l≤nj under the pinching assumption by [33, Lemma 2]. Hence by Theorem 3.7, (4.1) and (4.2), there exist infinitely many
(
N
,
m
1
,
1
,
…
,
m
1
,
n
1
,
…
,
m
p
,
1
,
…
,
m
p
,
n
p
)
=
(
N
,
m
1
,
…
,
m
1
,
…
,
m
p
,
…
,
m
p
)
∈
ℕ
q
+
1
such that
(4.4)
i
(
c
~
j
,
l
2
m
j
)
≥
2
N
-
e
(
P
c
~
j
,
l
)
2
≥
2
N
-
(
n
-
1
)
,
(4.5)
i
(
c
~
j
,
l
2
m
j
)
+
ν
(
c
~
j
,
l
2
m
j
)
≤
2
N
+
e
(
P
c
~
j
,
l
)
2
≤
2
N
+
(
n
-
1
)
,
(4.6)
i
(
c
~
j
,
l
2
m
j
-
m
)
+
ν
(
c
~
j
,
l
2
m
j
-
m
)
≤
2
N
-
(
i
(
c
~
j
,
l
)
+
2
S
P
c
~
j
,
l
+
(
1
)
-
ν
(
c
~
j
,
l
)
)
for all
m
∈
ℕ
,
(4.7)
i
(
c
~
j
,
l
2
m
j
+
m
)
≥
2
N
+
i
(
c
~
j
,
l
)
for all
m
∈
ℕ
for 1≤j≤p and 1≤l≤nj. Moreover, mjθπ∈ℤ, whenever e-1θ∈σ(Pc~j,l) and θπ∈ℚ. In fact, the m>1 cases in (4.6) and (4.7) follow from (4.3), other parts follow from Theorem 3.7 directly. More precisely, by Theorem 3.7,
(4.8)
m
j
=
(
[
N
M
i
^
(
c
~
j
,
l
)
]
+
χ
j
)
M
,
1
≤
j
≤
p
,
where χj=0 or 1 for 1≤j≤p and M∈ℕ such that Mθπ∈ℤ, whenever e-1θ∈σ(Pc~j,l) and θπ∈ℚ for some 1≤j≤p.
By Theorem 3.5, there exists a path fj∈C([0,1],Ω0(Pc~j,l)) such that fj(0)=Pc~j,l and
(4.9)
f
j
(
1
)
=
N
1
(
1
,
1
)
⋄
p
j
,
-
⋄
I
2
⋄
p
j
,
0
⋄
N
1
(
1
,
-
1
)
⋄
p
j
,
+
⋄
G
j
,
1
≤
j
≤
p
for some non-negative integers pj,-, pj,0, pj,+, and some symplectic matrix Gj satisfying 1∉σ(Gj). By (4.9) and Lemma 3.4 we obtain
(4.10)
2
S
P
c
~
j
,
l
+
(
1
)
-
ν
1
(
P
c
~
j
,
l
)
=
p
j
,
-
-
p
j
,
+
≥
-
p
j
,
+
≥
1
-
n
,
1
≤
j
≤
p
,
1
≤
l
≤
n
j
.
Using (4.2) and (4.10), estimates (4.4)–(4.7) become
(4.11)
i
(
c
~
j
,
l
2
m
j
)
≥
2
N
-
(
n
-
1
)
,
(4.12)
i
(
c
~
j
,
l
2
m
j
)
+
ν
(
c
~
j
,
l
2
m
j
)
≤
2
N
+
(
n
-
1
)
,
(4.13)
i
(
c
~
j
,
l
2
m
j
-
m
)
+
ν
(
c
~
j
,
l
2
m
j
-
m
)
≤
2
N
for all
m
∈
ℕ
,
(4.14)
i
(
c
~
j
,
l
2
m
j
+
m
)
≥
2
N
+
(
n
-
1
)
for all
m
∈
ℕ
for 1≤j≤p and 1≤l≤nj.
In order to prove Theorem 1.2, we need the following:
Lemma 4.1.
There exists j0∈{1,…,p} such that
i
(
c
~
j
0
,
l
2
m
j
0
)
+
ν
(
c
~
j
0
,
l
2
m
j
0
)
=
2
N
+
(
n
-
1
)
.
Moreover, C¯2N+n-1(E,c~j0,l2mj0)≠0 for 1≤l≤nj0.
Proof.
We prove by contradiction, i.e., suppose that
(4.15)
C
¯
2
N
+
n
-
1
(
E
,
c
~
j
,
l
2
m
j
)
=
0
,
1
≤
j
≤
p
,
1
≤
l
≤
n
j
.
Now (4.5)–(4.7) becomes
(4.16)
i
(
c
~
j
,
l
m
)
+
ν
(
c
~
j
,
l
m
)
≤
2
N
for all
m
<
2
m
j
,
(4.17)
i
(
c
~
j
,
l
2
m
j
)
+
ν
(
c
~
j
,
l
2
m
j
)
≤
2
N
+
n
-
1
,
(4.18)
i
(
c
~
j
,
l
m
)
≥
2
N
+
n
-
1
for all
m
>
2
m
j
.
By Theorem 3.7, we can choose N and {χj}1≤j≤p such that
(4.19)
|
N
M
i
^
(
c
~
j
,
l
)
-
[
N
M
i
^
(
c
~
j
,
l
)
]
-
χ
j
|
<
ϵ
<
1
1
+
∑
1
≤
j
≤
p
,
1
≤
l
≤
n
j
4
M
|
χ
^
(
c
~
j
,
l
)
|
,
1
≤
j
≤
p
.
By Theorem 2.5 we have
(4.20)
∑
1
≤
j
≤
p
,
1
≤
l
≤
n
j
χ
^
(
c
~
j
,
l
)
i
^
(
c
~
j
,
l
)
=
B
(
n
,
1
)
∈
ℚ
.
Note by Theorem 3.7, we can require that N∈ℕ further satisfies
(4.21)
2
N
B
(
n
,
1
)
∈
ℤ
.
Multiplying both sides of (4.20) by 2N yields
(4.22)
∑
1
≤
j
≤
p
,
1
≤
l
≤
n
j
2
N
χ
^
(
c
~
j
,
l
)
i
^
(
c
~
j
,
l
)
=
2
N
B
(
n
,
1
)
.
Claim 1.
We have
(4.23)
∑
1
≤
j
≤
p
,
1
≤
l
≤
n
j
2
m
j
χ
^
(
c
~
j
,
l
)
=
2
N
B
(
n
,
1
)
.
In fact, by (4.22), we have
2
N
B
(
n
,
1
)
=
∑
1
≤
j
≤
p
,
1
≤
l
≤
n
j
2
N
χ
^
(
c
~
j
,
l
)
i
^
(
c
~
j
,
l
)
=
∑
1
≤
j
≤
p
,
1
≤
l
≤
n
j
2
χ
^
(
c
~
j
,
l
)
(
[
N
M
i
^
(
c
~
j
,
l
)
]
+
χ
j
)
M
+
∑
1
≤
j
≤
p
,
1
≤
l
≤
n
j
2
χ
^
(
c
~
j
,
l
)
(
N
M
i
^
(
c
~
j
,
l
)
-
[
N
M
i
^
(
c
~
j
,
l
)
]
-
χ
j
)
M
(4.24)
≡
∑
1
≤
j
≤
p
,
1
≤
l
≤
n
j
2
m
j
χ
^
(
c
~
j
,
l
)
+
∑
1
≤
j
≤
p
,
1
≤
l
≤
n
j
2
M
χ
^
(
c
~
j
,
l
)
ϵ
j
.
By Proposition 2.3 and our choice of M, we have
(4.25)
2
m
j
T
(
c
~
j
,
l
)
∈
ℕ
,
1
≤
j
≤
p
,
1
≤
l
≤
n
j
.
Hence (2.3) implies that
(4.26)
2
m
j
χ
^
(
c
~
j
,
l
)
∈
ℤ
,
1
≤
j
≤
p
,
1
≤
l
≤
n
j
.
Now Claim 1 follows by (4.19), (4.21), (4.24) and (4.26).
Claim 2.
We have
(4.27)
∑
1
≤
j
≤
p
,
1
≤
l
≤
n
j
2
m
j
χ
^
(
c
~
j
,
l
)
=
M
0
-
M
1
+
M
2
-
⋯
+
(
-
1
)
2
N
+
n
-
2
M
2
N
+
n
-
2
.
In fact, by definition, the right-hand side of (4.27) is
M
0
-
M
1
+
M
2
-
⋯
+
(
-
1
)
2
N
+
n
-
2
M
2
N
+
n
-
2
=
∑
q
≤
2
N
+
n
-
2
,
m
≥
1
1
≤
j
≤
p
,
1
≤
l
≤
n
j
(
-
1
)
q
dim
C
¯
q
(
E
,
c
~
j
,
l
m
)
.
By (4.15)–(4.18) and Proposition 2.1, we have
M
0
-
M
1
+
M
2
-
⋯
+
(
-
1
)
2
N
+
n
-
2
M
2
N
+
n
-
2
=
∑
q
≤
2
N
+
n
-
2
,
1
≤
m
≤
2
m
j
1
≤
j
≤
p
,
1
≤
l
≤
n
j
(
-
1
)
q
dim
C
¯
q
(
E
,
c
~
j
,
l
m
)
,
=
∑
1
≤
m
≤
2
m
j
1
≤
j
≤
p
,
1
≤
l
≤
n
j
χ
(
c
~
j
,
l
m
)
,
where the second equality follows from (2.1), (4.15)–(4.17) and Proposition 2.1. By Proposition 2.3, (2.1)–(2.3) and (4.25), we have
∑
1
≤
m
≤
2
m
j
χ
(
c
~
j
,
l
m
)
=
∑
0
≤
s
<
2
m
j
/
T
(
c
~
j
,
l
)
1
≤
m
≤
T
(
c
~
j
,
l
)
χ
(
c
~
j
,
l
s
T
(
c
~
j
,
l
)
+
m
)
=
2
m
j
T
(
c
~
j
,
l
)
∑
1
≤
m
≤
T
(
c
~
j
,
l
)
χ
(
c
~
j
,
l
m
)
=
2
m
j
χ
^
(
c
~
j
,
l
)
.
This proves Claim 2.
In order to prove the lemma, we consider the following two cases according to the parity of n.
Case 1: n=2k+1 is odd. In this case, we have by (2.4),
(4.28)
B
(
n
,
1
)
=
n
+
1
2
(
n
-
1
)
=
k
+
1
2
k
.
By Theorem 3.7 we may further assume N=mk for some m∈ℕ. Thus by (4.23), (4.27) and (4.28), we have
M
0
-
M
1
+
M
2
-
⋯
+
(
-
1
)
2
N
+
n
-
2
M
2
N
+
n
-
2
=
m
(
k
+
1
)
.
On the other hand, we have by (2.5)
b
0
-
b
1
+
b
2
-
⋯
+
(
-
1
)
2
N
+
n
-
2
b
2
N
+
n
-
2
=
b
2
k
+
(
b
2
k
+
2
+
⋯
+
b
4
k
+
⋯
+
b
2
m
k
+
2
+
⋯
+
b
2
m
k
+
2
k
)
-
b
2
m
k
+
2
k
=
1
+
m
(
k
-
1
+
2
)
-
2
(4.29)
=
m
(
k
+
1
)
-
1
.
In fact, we cut off the sequence {b2k+2,…,b2mk+2k} into m pieces, each of them contains k terms. Moreover, each piece contains 1 for k-1 times and 2 for one time. Thus (4.29) holds. Now by Theorem 2.7 and (4.29), we have
-
m
(
k
+
1
)
=
M
2
N
+
n
-
2
-
M
2
N
+
n
-
3
+
⋯
+
M
1
-
M
0
≥
b
2
N
+
n
-
2
-
b
2
N
+
n
-
3
+
⋯
+
b
1
-
b
0
=
-
(
m
(
k
+
1
)
-
1
)
.
This contradiction yields the lemma for n being odd.
Case 2: n=2k is even. In this case, we have by (2.4),
(4.30)
B
(
n
,
1
)
=
-
n
2
n
-
2
=
-
k
2
k
-
1
.
As in Case 1, we may assume N=m(2k-1) for some m∈ℕ. Thus by (4.23), (4.27) and (4.30), we have
(4.31)
M
0
-
M
1
+
M
2
-
⋯
+
(
-
1
)
2
N
+
n
-
2
M
2
N
+
n
-
2
=
-
2
m
k
.
On the other hand, we have by (2.6)
b
0
-
b
1
+
b
2
-
⋯
+
(
-
1
)
2
N
+
n
-
2
b
2
N
+
n
-
2
=
-
b
2
k
-
1
-
(
b
2
k
+
1
+
⋯
+
b
6
k
-
3
+
⋯
+
b
(
m
-
1
)
(
4
k
-
2
)
+
2
k
+
1
+
⋯
+
b
m
(
4
k
-
2
)
+
2
k
-
1
)
+
b
m
(
4
k
-
2
)
+
2
k
-
1
=
-
1
-
m
(
2
k
-
2
+
2
)
+
2
(4.32)
=
-
2
m
k
+
1
.
In fact, we cut off the sequence {b2k+1,…,bm(4k-2)+2k-1} into m pieces, each of them contains 2k-1 terms. Moreover, each piece contains 1 for 2k-2 times and 2 for one time. Thus (4.32) holds. Now by (4.31), (4.32) and Theorem 2.7, we have
-
2
m
k
=
M
2
N
+
n
-
2
-
M
2
N
+
n
-
3
+
⋯
+
M
1
-
M
0
≥
b
2
N
+
n
-
2
-
b
2
N
+
n
-
3
+
⋯
+
b
1
-
b
0
=
-
2
m
k
+
1
.
This contradiction yields the lemma for n being even. ∎
We will use the following theorem of N. Hingston. Note that the proof of Hingston’s theorem does not use the special properties of Riemannian metric, hence it holds for Finsler metric as well.
Theorem 4.2 (follows from [11, Proposition 1], cf. [14, Lemma 3.4.12] and [15]).
Let c be a closed geodesic of length L on a compact Finsler manifold (M,F) such that as a critical orbit of the energy functional E on ΛM, every orbit S1⋅cm of its iteration cm is isolated. Suppose
(4.33)
i
(
c
m
)
+
ν
(
c
m
)
≤
m
(
i
(
c
)
+
ν
(
c
)
)
-
(
n
-
1
)
(
m
-
1
)
for all
m
∈
ℕ
,
(4.34)
k
ν
(
c
)
(
c
)
≠
0
.
Then (M,F) has infinitely many prime closed geodesics.
Note that in (4.34), we have used the Shifting theorem in [10]. Especially, (4.34) means that c is a local maximum in the local characteristic manifold Nc at c.
Lemma 4.3.
The closed geodesic c~j0,l found in Lemma 4.1 satisfy the following:
e
(
P
c
~
j
0
,
l
)
=
2
n
-
2
, i.e.,
c
~
j
0
,
l
is elliptic.
P
c
~
j
0
,
l
does not contain
N
1
(
1
,
1
)
, N1(-1,-1) and non-trivial N2(ω,b).
Any trivial
N
2
(
ω
,
b
)
contained in
P
c
~
j
0
,
l
must satisfies
θ
π
∈
ℚ
, where
ω
=
e
-
1
θ
.
We have
k
ν
(
c
~
j
0
,
l
T
(
c
~
j
0
,
l
)
)
(
c
~
j
0
,
l
T
(
c
~
j
0
,
l
)
)
≠
0
.
Hence
c
~
j
0
,
l
T
(
c
~
j
0
,
l
)
is a local maximum of the energy functional in the local characteristic manifold at
c
~
j
0
,
l
T
(
c
~
j
0
,
l
)
.
P
c
~
j
0
,
l
must contain a term
R
(
θ
)
with
θ
π
∉
ℚ
.
Proof.
Note that by (3.11), Lemma 4.1 and Theorem 3.6, we have
2
N
+
(
n
-
1
)
=
i
(
c
~
j
0
,
l
2
m
j
0
)
+
ν
(
c
~
j
0
,
l
2
m
j
0
)
(4.35)
≤
-
i
(
c
~
j
0
,
l
2
m
j
0
+
1
)
-
i
(
c
~
j
0
,
l
)
+
e
(
P
c
~
j
0
,
l
)
2
(4.36)
=
2
N
+
e
(
P
c
~
j
0
,
l
)
2
≤
2
N
+
(
n
-
1
)
.
Hence (i) holds. If any one of (ii)–(iii) does not hold, then by Theorem 3.5, the strict inequality in (4.35) must hold and yields a contraction.
(iv) This follows directly from Proposition 2.3 and Lemma 4.1.
We prove (v) by contradiction. Consider
g
l
=
c
~
j
0
,
l
T
(
c
~
j
0
,
l
)
.
By (i)–(iii) and the assumption, Pgl can be connected in Ω0(Pgl) to I2p0′⋄N1(1,-1)⋄p+′ with p0′+p+′=n-1 as in Theorem 3.5. In fact, by (i)–(iii) and the assumption, the basic normal form decomposition (3.4) in Theorem 3.5 becomes
P
c
~
j
0
,
l
=
I
2
p
0
⋄
N
1
(
1
,
-
1
)
⋄
p
+
⋄
N
1
(
-
1
,
1
)
⋄
q
-
⋄
(
-
I
2
q
0
)
⋄
R
(
θ
1
)
⋄
⋯
⋄
R
(
θ
r
)
⋄
N
2
(
λ
1
,
v
1
)
⋄
⋯
⋄
N
2
(
λ
r
0
,
v
r
0
)
together with θjπ∈ℚ for 1≤j≤r, βjπ∈ℚ for 1≤j≤r0 and p0+p++q-+q0+r+2r0=n-1. By Proposition 2.3, we have 12πT(c~j0,l)θj∈ℤ, 12πT(c~j0,l)βj∈ℤ and 2∣T(c~j0,l) whenever -1∈σ(Pc~j0,l). Hence we obtain R(θj)T(c~j0,l)=I2, N2(λj,vj)T(c~j0,l) can be connected within Ω0(N2(λj,vj)T(c~j0,l)) to N1(1,-1)⋄2 and (-I2)T(c~j0,l)=I2, and N1(-1,1)T(c~j0,l) can be connected within Ω0(N1(-1,1)T(c~j0,l)) to N1(1,-1) whenever -1∈σ(Pc~j0,l). Thus p0′=p0+q0+r and p+′=p++q-+2r0 and then Pgl behaves as claimed.
Now by Theorem 3.5, we have
i
(
g
l
m
)
=
m
(
i
(
g
l
)
+
p
0
′
)
-
p
0
′
,
ν
(
g
l
m
)
≡
2
p
0
′
+
p
+
′
for all
m
∈
ℕ
.
Hence
i
(
g
l
m
)
+
ν
(
g
l
m
)
=
m
(
i
(
g
l
)
+
p
0
′
)
+
p
0
′
+
p
+
′
for all
m
∈
ℕ
.
On the other hand
m
(
i
(
g
l
)
+
ν
(
g
l
)
)
-
(
n
-
1
)
(
m
-
1
)
=
m
(
i
(
g
l
)
+
2
p
0
′
+
p
+
′
)
-
(
p
0
′
+
p
+
′
)
(
m
-
1
)
=
m
(
i
(
g
l
)
+
p
0
′
)
+
p
0
′
+
p
+
′
for all
m
∈
ℕ
.
By (iv), we have
k
ν
(
g
l
)
(
g
l
)
=
k
ν
(
c
~
j
0
,
l
T
(
c
~
j
0
,
l
)
)
(
c
~
j
0
,
l
T
(
c
~
j
0
,
l
)
)
≠
0
.
Hence we can use Theorem 4.2 to obtain infinitely many prime closed geodesics, which contradicts Assumption (F). This completes the proof of Lemma 4.3. ∎
Lemma 4.4.
There exists no closed geodesic g on (M~,F~) such that Pg=N1(1,-1)⋄(n-1) and kν(g)(g)≠0.
Proof.
By the proof of Lemma 4.3, we can use Theorem 4.2 to obtain infinitely many prime closed geodesics, which contradicts Assumption (F). ∎
Proof of Theorem 1.2.
By Lemma 2.10, for every i∈ℕ, there exist some m(i),j(i)∈ℕ such that
(4.37)
E
(
c
~
j
(
i
)
,
l
m
(
i
)
)
=
κ
i
,
C
¯
2
i
+
dim
(
z
)
-
2
(
E
,
c
~
j
(
i
)
,
l
m
(
i
)
)
≠
0
,
and by Section 2, we have dim(z)=n+1.
Claim 1.
We have the following:
(4.38)
m
(
i
)
=
2
m
j
(
i
)
if
2
i
+
dim
(
z
)
-
2
∈
(
2
N
,
2
N
+
n
-
1
)
.
In fact, we have
C
¯
q
(
E
,
c
~
j
,
l
m
)
=
0
if
q
∈
(
2
N
,
2
N
+
n
-
1
)
for 1≤j≤p, 1≤l≤nj and m≠2mj by (4.13), (4.14) and Proposition 2.1. Thus in order to satisfy (4.37), we must have m(i)=2mj(i).
By Lemma 2.9, for 2α+dim(z)-2,2β+dim(z)-2∈(2N,2N+n-1) with α≠β,
(4.39)
E
(
c
~
j
(
α
)
,
l
2
m
j
(
α
)
)
=
κ
α
≠
κ
β
=
E
(
c
~
j
(
β
)
,
k
2
m
j
(
β
)
)
.
Thus c~j(α),l≠c~j(β),k for 1≤l≤nj(α) and 1≤k≤nj(β). Therefore there are
{
i
:
2
i
+
dim
(
z
)
-
2
∈
(
2
N
,
2
N
+
n
-
1
)
}
#
=
[
n
2
]
-
1
closed geodesics on (M,F). By a permutation of {1,…,p}, we may denote these closed geodesics by {c1,…,c[n2]-1}.
Claim 2.
If n is odd, then it is impossible that C¯2N(E,c~j,l2mj-m)≠0 for some 1≤j≤p and m∈N.
Suppose the contrary; then by (4.3), (4.6), (4.10) and Proposition 2.1, we have
i
(
c
~
j
,
l
2
m
j
-
1
)
+
ν
(
c
~
j
,
l
2
m
j
-
1
)
=
2
N
,
P
c
~
j
,
l
=
N
1
(
1
,
-
1
)
⋄
(
n
-
1
)
together with i(c~j,l)=n-1 for some 1≤j≤p. Thus by Proposition 2.3, we have
k
ν
(
c
~
j
,
l
)
(
c
~
j
,
l
)
≠
0
.
Hence we can use Lemma 4.4 to obtain infinitely many prime closed geodesics, which contradicts Assumption (F). This proves Claim 2.
Thus by Claim 2, (4.14) and Proposition 2.1, for n being odd and 2i+dim(z)-2=2N we have
(4.40)
E
(
c
~
j
(
i
)
,
l
2
m
j
(
i
)
)
=
κ
i
,
C
¯
2
N
(
E
,
c
~
j
(
i
)
,
l
2
m
j
(
i
)
)
≠
0
,
thus we have one more closed geodesic on (M,F) by Lemma 2.9. Hence we have [n+12]-1 closed geodesics on (M,F). We may denote these closed geodesics by {c1,…,c[n+12]-1}.
By Lemma 4.1, there exists j0∈{1,…,p} such that
i
(
c
~
j
0
,
l
2
m
j
0
)
+
ν
(
c
~
j
0
,
l
2
m
j
0
)
=
2
N
+
(
n
-
1
)
;
moreover, C¯2N+n-1(E,c~j0,l2mj0)≠0 for 1≤l≤nj0. By Proposition 2.2, we have
(4.41)
C
¯
q
(
E
,
c
~
j
0
,
l
2
m
j
0
)
=
0
,
q
≠
2
N
+
(
n
-
1
)
,
1
≤
l
≤
n
j
0
.
By (4.37), (4.40) and (4.41), cj0∉{c1,…,c[n+12]-1}. Hence we have [n+12] closed geodesics on (M,F). The proof of Theorem 1.2 is complete. ∎
Proof of Theorem 1.3.
By [17, Theorem 1.1] or [20, Lemma 3.2], cj is hyperbolic if and only if c~j,l are hyperbolic for 1≤l≤nj. By (4.4) and (4.5), a hyperbolic closed geodesic c~j,l must satisfy
i
(
c
~
j
,
l
2
m
j
)
=
2
N
=
i
(
c
~
j
,
l
2
m
j
)
+
ν
(
c
~
j
,
l
2
m
j
)
.
Thus by Proposition 2.1 we have
(4.42)
C
¯
q
(
E
,
c
~
j
,
l
2
m
j
)
=
0
,
q
≠
2
N
.
By (4.37) and (4.42), the closed geodesics {c1,…,c[n2]-1} are non-hyperbolic. By Lemma 4.1 and (4.42), the closed geodesic cj0 is non-hyperbolic. Therefore there are [n2] non-hyperbolic closed geodesics on (M,F). ∎
Proof of Theorem 1.4.
By [17, Theorem 1.1] or [20, Lemma 3.2], Pcj possess an eigenvalue of the form exp(πiμ) with an irrational μ if and only if Pc~j,l have the same property for 1≤l≤nj.
Firstly we prove the following:
Claim 1.
There are at least [n-12] closed geodesics cjk for 1≤k≤[n-12] on (M,F) such that c~jk,l∈V∞(M~,F~) for 1≤k≤[n-12] and 1≤l≤njk.
As in the proof of Theorem 1.2, for any N chosen in (4.11)–(4.14) fixed and 2i+dim(z)-2∈[2N,2N+n-1), there exist some 1≤j(i)≤p such that c~j(i) is (2mj(i),i)-variationally visible by (4.37), (4.38) and (4.40). Moreover, if i1≠i2, then we must have j(i1)≠j(i2) by (4.39), (4.40) and Lemma 2.9. Hence the map
(4.43)
Ψ
:
(
2
ℕ
+
dim
(
z
)
-
2
)
∩
[
2
N
,
2
N
+
n
-
1
)
→
{
c
j
}
1
≤
j
≤
p
,
2
i
+
dim
(
z
)
-
2
↦
c
j
(
i
)
is injective. We remark here that if there are more that one cj satisfy (4.37), we take any one of it. Since we have infinitely many N satisfying (4.11)–(4.14) and the number of prime closed geodesics is finite, Claim 1 must hold.
Claim 2.
Among the [n-12] closed geodesics {cjk}1≤k≤[n-12] in Claim 1, there are at least [n-12]-1 ones possessing irrational average indices.
We prove the claim as the following: By Theorem 3.7, we can obtain infinitely many N in (4.11)–(4.14) satisfying the further properties
(4.44)
N
M
i
^
(
c
~
j
,
l
)
∈
ℕ
and
χ
j
=
0
if
i
^
(
c
~
j
,
l
)
∈
ℚ
.
Now suppose i^(c~jk1,l)∈ℚ and i^(c~jk2,l)∈ℚ hold for some distinct 1≤k1,k2≤[n-12]-1. Then by (4.8) and (4.44) we have
2
m
j
k
1
i
^
(
c
~
j
k
1
,
l
)
=
2
(
[
N
M
i
^
(
c
~
j
k
1
,
l
)
]
+
χ
j
k
1
)
M
i
^
(
c
~
j
k
1
,
l
)
=
2
(
N
M
i
^
(
c
~
j
k
1
,
l
)
)
M
i
^
(
c
~
j
k
1
,
l
)
=
2
N
=
2
(
N
M
i
^
(
c
~
j
k
2
,
l
)
)
M
i
^
(
c
~
j
k
2
,
l
)
(4.45)
=
2
(
[
N
M
i
^
(
c
~
j
k
2
,
l
)
]
+
χ
j
k
2
)
M
i
^
(
c
~
j
k
2
,
l
)
=
2
m
j
k
2
i
^
(
c
~
j
k
2
,
l
)
.
On the other hand, by (4.43), we have
Ψ
(
2
i
1
+
dim
(
z
)
-
2
)
=
c
j
k
1
,
Ψ
(
2
i
2
+
dim
(
z
)
-
2
)
=
c
j
k
2
for some
i
1
≠
i
2
.
Thus by (4.37), (4.38), (4.40) and Lemma 2.9, we have
E
(
c
~
j
k
1
,
l
2
m
j
k
1
)
=
κ
i
1
≠
κ
i
2
=
E
(
c
~
j
k
2
,
l
2
m
j
k
2
)
.
Since c~jk1,l,c~jk2,l∈𝒱∞(M~,F~), by Theorem 2.12 we have
i
^
(
c
~
j
k
1
,
l
)
L
(
c
~
j
k
1
,
l
)
=
2
σ
=
i
^
(
c
~
j
k
2
,
l
)
L
(
c
~
j
k
2
,
l
)
.
Note that we have the relations
L
(
c
~
m
)
=
m
L
(
c
~
)
,
i
^
(
c
~
m
)
=
m
i
^
(
c
~
)
,
L
(
c
~
)
=
2
E
(
c
~
)
for all
m
∈
ℕ
for any closed geodesic c~ on (M~,F~). Hence we have
2
m
j
k
1
i
^
(
c
~
j
k
1
,
l
)
=
2
σ
⋅
2
m
j
k
1
L
(
c
~
j
k
1
,
l
)
=
2
σ
L
(
c
~
j
k
1
,
l
2
m
j
k
1
)
=
2
σ
2
E
(
c
~
j
k
1
,
l
2
m
j
k
1
)
=
2
σ
2
κ
i
1
≠
2
σ
2
κ
i
2
=
2
σ
2
E
(
c
~
j
k
2
,
l
2
m
j
k
2
)
=
2
σ
L
(
c
~
j
k
2
,
l
2
m
j
k
2
)
=
2
σ
⋅
2
m
j
k
2
L
(
c
~
j
k
2
,
l
)
=
2
m
j
k
2
i
^
(
c
~
j
k
2
,
l
)
.
This contradicts (4.45) and then we must have i^(c~jk1,l)∈ℝ∖ℚ or i^(c~jk2,l)∈ℝ∖ℚ. Hence there is at most one 1≤k≤[n-12] such that i^(c~jk,l)∈ℚ, i.e., there are at least [n-12]-1 ones possessing irrational average indices. This proves Claim 2.
Suppose that cjk is any closed geodesic found in Claim 2; then we have i^(c~jk,l)∈ℝ∖ℚ. Therefore by Theorem 3.5, the linearized Poincaré map Pc~jk,l of c~jk,l must contain a term R(θ) with θπ∉ℚ. By Lemma 4.3, the closed geodesic c~j0,l also has this property. Moreover, we have cj0∉{cjk}1≤k≤[n-12] as in the proof of Theorem 1.2. Thus Theorem 1.4 is true. ∎
Proof of Theorem 1.5.
We prove the theorem by contradiction, i.e., we assume that there are exactly two closed geodesics c1 and c2 on (M,F) by Theorem 1.2.
Step 1.
We can write
(4.46)
P
c
~
j
,
l
=
R
(
θ
j
)
⋄
M
j
′
,
1
≤
j
≤
2
,
1
≤
l
≤
n
j
,
with θjπ∉Q and Mj′∈Sp(2).
By Lemma 4.3, we may assume that Pc~1,l contains a term R(θ1) with θ1π∉ℚ. and
k
ν
(
c
~
1
,
l
T
(
c
~
1
,
l
)
)
(
c
~
1
,
l
T
(
c
~
1
,
l
)
)
≠
0
.
Thus in order to prove Step 1, we only need to prove (4.46) for j=2. By Theorem 3.5, we have
i
^
(
c
~
1
,
l
)
=
i
(
c
~
1
,
1
)
+
p
-
+
p
0
-
r
+
∑
1
≤
j
≤
r
ϕ
j
π
for some p-,p0,r∈ℕ0. By Lemma 4.3, we have r≥1 and then
i
^
(
c
~
1
,
l
)
=
i
(
c
~
1
,
1
)
+
p
-
+
p
0
-
r
+
∑
2
≤
j
≤
r
ϕ
j
π
+
θ
1
π
≡
Δ
+
θ
1
π
.
and
α
1
,
1
i
^
(
c
~
1
,
l
)
=
θ
1
π
Δ
+
θ
1
π
,
where we use notations as in Theorem 3.7, i.e., α1,1=θ1π. We have the following three cases.
Case 1.1: We have Δ∈Q. Denote β=Δ+θ1π∉ℚ. Then we have
(
1
M
i
^
(
c
~
1
,
l
)
,
α
1
,
1
i
^
(
c
~
1
,
l
)
)
=
(
1
M
β
,
1
-
Δ
β
)
.
Thus if NMi^(c~1,l)=K+μ for some K∈ℤ and μ∈(-1,1), we have
N
α
1
,
1
i
^
(
c
~
1
,
l
)
=
N
-
M
Δ
K
-
M
Δ
μ
.
Note that by the choice of M, we have MΔ∈ℤ, therefore by (3.15) and (3.16), we have
{
χ
1
,
1
=
1
if
χ
1
=
0
,
χ
1
,
1
=
0
if
χ
1
=
1
.
Thus either (χ1,χ1,1)=(1,0) or (χ1,χ1,1)=(0,1) holds. By [25, (4.16) and (4.17)], we have
{
m
1
α
1
,
1
}
=
{
{
N
α
1
,
1
i
^
(
c
~
1
,
l
)
}
-
χ
1
,
1
+
(
χ
1
-
{
N
M
i
^
(
c
~
1
,
l
)
}
)
M
α
1
,
1
}
=
{
A
1
,
1
(
N
)
+
B
1
,
1
(
N
)
}
=
{
{
{
N
α
1
,
1
i
^
(
c
~
1
,
l
)
}
-
χ
1
,
1
+
(
χ
1
-
{
N
M
i
^
(
c
~
1
,
l
)
}
)
M
α
1
,
1
}
if
(
χ
1
,
χ
1
,
1
)
=
(
1
,
0
)
,
{
1
+
{
N
α
1
,
1
i
^
(
c
~
1
,
l
)
}
-
χ
1
,
1
+
(
χ
1
-
{
N
M
i
^
(
c
~
1
,
l
)
}
)
M
α
1
,
1
}
if
(
χ
1
,
χ
1
,
1
)
=
(
0
,
1
)
,
where
A
1
,
1
(
N
)
=
{
N
α
1
,
1
i
^
(
c
~
1
,
l
)
}
-
χ
1
,
1
and
B
1
,
1
(
N
)
=
(
χ
1
-
{
N
M
i
^
(
c
~
1
,
l
)
}
)
M
α
1
,
1
.
In fact, we have
A
1
,
1
(
T
)
>
0
,
B
1
,
1
(
T
)
>
0
for
(
χ
1
,
χ
1
,
1
)
=
(
1
,
0
)
and
A
1
,
1
(
T
)
<
0
,
B
1
,
1
(
T
)
<
0
for
(
χ
1
,
χ
1
,
1
)
=
(
0
,
1
)
,
thus the last equality above holds. Hence by (3.16), we have
{
{
m
1
α
1
,
1
}
<
(
2
M
+
1
)
ϵ
if
(
χ
1
,
χ
1
,
1
)
=
(
1
,
0
)
,
{
m
1
α
1
,
1
}
>
1
-
(
2
M
+
1
)
ϵ
if
(
χ
1
,
χ
1
,
1
)
=
(
0
,
1
)
,
where we have used the fact that α1,1=θ1π∈(0,2). Choosing ϵ∈(0,12M+1min{θ12π,1-θ12π}), by Theorem 3.5, we have
i
(
c
~
1
,
l
2
m
1
+
1
)
-
i
(
c
~
1
,
l
2
m
1
)
=
i
(
c
~
1
,
l
)
+
p
-
+
p
0
-
r
+
∑
1
≤
i
≤
r
(
2
E
(
(
2
m
1
+
1
)
ϕ
i
2
π
)
-
2
E
(
2
m
1
ϕ
i
2
π
)
)
+
q
0
+
q
+
=
2
E
(
(
2
m
1
+
1
)
θ
1
2
π
)
-
2
E
(
2
m
1
θ
1
2
π
)
+
i
(
c
~
1
,
l
)
+
p
-
+
p
0
-
r
+
2
(
r
-
1
)
+
q
0
+
q
+
≡
2
E
(
(
2
m
1
+
1
)
θ
1
2
π
)
-
2
E
(
2
m
1
θ
1
2
π
)
+
Π
=
2
(
E
(
2
m
1
θ
1
2
π
+
θ
1
2
π
)
-
E
(
2
m
1
θ
1
2
π
)
)
+
Π
=
2
(
E
(
{
m
1
α
1
,
1
}
+
θ
1
2
π
)
-
E
(
{
m
1
α
1
,
1
}
)
)
+
Π
(4.47)
=
{
Π
if
{
m
1
α
1
,
1
}
<
(
2
M
+
1
)
ϵ
,
2
+
Π
if
{
m
1
α
1
,
1
}
>
1
-
(
2
M
+
1
)
ϵ
,
where p-,p0,q0,q+∈ℕ0, r∈ℕ, ϕ1=θ1 and ϕ22π∈(0,1)∩ℚ, m1ϕ2π∈ℤ whenever r=2. Hence by Theorems 3.7–3.9, we can choose another N′∈ℕ and
(4.48)
m
j
′
=
(
[
N
′
M
i
^
(
c
~
j
,
l
)
]
+
χ
j
′
)
M
,
1
≤
j
≤
2
,
such that {m1α1,1}<(2M+1)ϵ and {m1′α1,1}>1-(2M+1)ϵ. By (3.11), (4.47) and ν(c~1,l2m1)=ν(c~1,l2m1′), we have
i
(
c
~
1
,
l
2
m
1
′
)
+
ν
(
c
~
1
,
l
2
m
1
′
)
≤
2
N
′
+
(
n
-
1
)
-
2
.
Hence the closed geodesic found in Lemma 4.1 for N′ must be c~2,l by Proposition 2.1, and then Step 1 holds in this case by Lemma 4.3.
Case 1.2: We have Δ∉Q and i^(c~1,l)∈Q. In this case we have Pc~1,l=R(ϕ1)⋄R(ϕ2) with ϕiπ∉ℚ for i=1,2 and ϕ1=θ1. Note that χ1=0 and {NMi^(c~1,l)}=0 by Theorem 3.7. Let α1,i=ϕiπ. Therefore we have
{
m
1
α
1
,
i
}
=
{
A
1
,
i
(
N
)
}
=
{
{
N
α
1
,
i
i
^
(
c
~
1
,
l
)
}
-
χ
1
,
i
}
.
Hence
{
{
m
1
α
1
,
i
}
<
ϵ
if
(
χ
1
,
χ
1
,
i
)
=
(
0
,
0
)
,
{
m
1
α
1
,
i
}
>
1
-
ϵ
if
(
χ
1
,
χ
1
,
i
)
=
(
0
,
1
)
.
As in Case 1.1, we have by (3.11), (4.36) and Theorem 3.5,
i
(
c
~
1
,
l
)
-
2
=
2
N
+
i
(
c
~
1
,
l
)
-
(
2
N
+
(
3
-
1
)
=
i
(
c
~
1
,
l
2
m
1
+
1
)
-
i
(
c
~
1
,
l
2
m
1
)
(4.49)
=
i
(
c
~
1
,
l
)
-
2
+
∑
1
≤
i
≤
2
(
2
E
(
(
2
m
1
+
1
)
ϕ
i
2
π
)
-
2
E
(
2
m
1
ϕ
i
2
π
)
)
.
This implies
E
(
(
2
m
1
+
1
)
ϕ
i
2
π
)
=
E
(
2
m
1
ϕ
i
2
π
)
for
i
=
1
,
2
,
and then {m1α1,i}<ϵ for i=1,2 must hold. As in Case 1.1, we can choose another N′∈ℕ with mi′ as defined in (4.48) such that {m1′α1,i}>1-ϵ for i=1,2. By (4.49), we have
i
(
c
~
1
,
l
2
m
1
′
)
=
i
(
c
~
1
,
l
2
m
1
′
+
1
)
-
(
i
(
c
~
1
,
l
)
-
2
+
4
)
=
2
N
′
+
i
(
c
~
1
,
l
)
-
i
(
c
~
1
,
l
)
-
2
=
2
N
′
-
2
.
Hence the closed geodesic found in Lemma 4.1 for N′ must be c~2,l, and then Step 1 holds in this case by Lemma 4.3.
Case 1.3: We have Δ∉Q and i^(c~1,l)∉Q. In this case Pc~1,l=R(ϕ1)⋄R(ϕ2) with ϕiπ∉ℚ for i=1,2 and ϕ1=θ1. By Theorem 3.5, we have T(c~1,l)=1 and then
χ
^
(
c
~
1
,
l
)
=
χ
(
c
~
1
,
l
)
=
k
ν
(
c
~
1
,
l
)
(
c
~
1
,
l
)
≠
0
by Proposition 2.1. By Theorem 2.5,
(4.50)
n
1
χ
^
(
c
~
1
,
1
)
i
^
(
c
~
1
,
1
)
+
n
2
χ
^
(
c
~
2
,
1
)
i
^
(
c
~
2
,
1
)
=
∑
1
≤
j
≤
2
,
1
≤
l
≤
n
j
χ
^
(
c
~
j
,
l
)
i
^
(
c
~
j
,
l
)
=
B
(
3
,
1
)
=
1
.
Thus we have i^(c~2,l)∉ℚ also by i^(c~1,l)∉ℚ and (4.50), and then Step 1 is true in his case by Theorem 3.5.
Step 2.
We have
M
k
=
b
k
=
{
1
if
q
=
2
,
2
if
q
∈
2
N
+
2
,
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
.
By Theorem 2.7, it is sufficient to show that M2k+1=0 for for all k∈ℤ, i.e.,
(4.51)
C
¯
2
k
+
1
(
E
,
c
~
j
,
l
m
)
=
0
for all
k
∈
ℤ
,
1
≤
j
≤
2
,
1
≤
l
≤
n
j
.
By Step 1, we can write Pc~j,l=R(θj)⋄Mj′ with θjπ∉ℚ and Mj′∈Sp(2) for j=1,2. By Lemma 4.1 and (4.40), we have
(4.52)
C
¯
q
(
E
,
c
j
,
l
2
m
j
)
≠
0
,
q
=
2
N
or
2
N
+
2
.
Note that by Theorem 3.5, the matrix I2 and N1(1,1) have the same index iteration formula (3.5) and can be viewed as R(θ) with θ=2π. Although I2 and N1(1,1) have different nullity iteration formula (3.6), the discussion below for I2 works also for N1(1,1). Thus in order to shorten the length of the paper, we will not discuss N1(1,1) separately. Similarly the matrix -I2 and N1(-1,1) have the same index iteration formula (3.5) and can be viewed as R(θ) with θ=π. With this point of view, we have the following classification by Theorem 3.5:
Case 2.1: Pc~j,l=R(θj)⋄R(φj) with φjπ∈(0,2]. By Theorem 3.5, we have
i
(
c
~
j
,
l
m
)
=
m
(
i
(
c
~
j
,
l
)
-
2
)
+
2
E
(
m
θ
j
2
π
)
+
2
E
(
m
φ
j
2
π
)
-
2
,
ν
(
c
~
j
,
l
m
)
=
2
(
1
-
φ
(
m
φ
j
2
π
)
)
for all
m
∈
ℕ
,
and i(c~j,l)∈2ℕ0. If φj2π∉ℚ, we have i(c~j,lm)∈2ℕ0 and ν(c~j,lm)=0 for m∈ℕ. Hence (4.51) holds by Proposition 2.1. If φj2π∈ℚ, write φj2π=rs with r,s∈ℕ and (r,s)=1. Then i(c~j,lm) is always even and ν(c~j,lm)=2 if s∣m, ν(c~j,lm)=0 otherwise. Thus by (4.52) and Proposition 2.1, k0(c~j,l2mj)≠0 or k2(c~j,l2mj)≠0, and then k1(c~j,l2mj)=0 by Proposition 2.2. By Proposition 2.3, we have
(4.53)
k
l
(
c
~
j
,
l
m
)
=
k
l
(
c
~
j
,
l
2
m
j
)
,
s
∣
m
,
for all
l
∈
ℤ
.
Hence (4.51) holds by Proposition 2.1 and (4.53).
Case 2.2: Pc~j,l=R(θj)⋄N1(1,-1). By Theorem 3.5, we have
i
(
c
~
j
,
l
m
)
=
m
(
i
(
c
~
j
,
l
)
-
1
)
+
2
E
(
m
θ
j
2
π
)
-
1
,
ν
(
c
~
j
,
l
m
)
=
1
for all
m
∈
ℕ
,
and i(c~j,l)∈2ℕ-1. Then i(c~j,lm) is always odd. Thus by (4.52) and Proposition 2.1, We have k1(c~j,l2mj)≠0 and then k0(c~j,l2mj)=0 by Proposition 2.2. By Proposition 2.3, we have
(4.54)
k
l
(
c
~
j
,
l
m
)
=
k
l
(
c
~
j
,
l
2
m
j
)
,
m
∈
ℕ
,
for all
l
∈
ℤ
.
Hence (4.51) holds by Proposition 2.1 and (4.54).
Case 2.3: Pc~j,l=R(θj)⋄N1(-1,1). By Theorem 3.5, we have
i
(
c
~
j
,
l
m
)
=
m
(
i
(
c
~
j
,
l
)
-
1
)
+
2
E
(
m
θ
j
2
π
)
-
1
,
ν
(
c
~
j
,
l
m
)
=
1
+
(
-
1
)
m
2
for all
m
∈
ℕ
,
and i(c~j,l)∈2ℕ0. Then i(c~j,lm)∈2ℕ0 and ν(c~j,lm)=0 if m∈2ℕ-1; i(c~j,lm)∈2ℕ-1 and ν(c~j,lm)=1 if m∈2ℕ. Thus by (4.52) and Proposition 2.1, k1(c~j,l2mj)≠0 and then k0(c~j,l2mj)=0 by Proposition 2.2. By Proposition 2.3, we have
(4.55)
k
l
(
c
~
j
,
l
m
)
=
k
l
(
c
~
j
,
l
2
m
j
)
,
m
∈
2
ℕ
,
for all
l
∈
ℤ
.
Hence (4.51) holds by Proposition 2.1 and (4.55).
Case 2.4: Pc~j,l=R(θj)⋄H with H being hyperbolic. By Theorem 3.5, we have
i
(
c
~
j
,
l
m
)
=
m
(
i
(
c
~
J
,
l
)
-
1
)
+
2
E
(
m
θ
j
2
π
)
-
1
,
ν
(
c
~
j
,
l
m
)
=
0
,
and i(c~j,l)∈ℕ0. If i(c~j,l)∈2ℕ-1, then i(c~j,lm)∈2ℕ-1 for m∈ℕ. This contradicts (4.52) by Proposition 2.1. Hence this case cannot appear. If i(c~j,l)∈2ℕ0, then i(c~j,lm)∈2ℕ0 if m∈2ℕ-1; i(c~j,lm)∈2ℕ-1 if m∈2ℕ. This contradicts (4.52) by Proposition 2.1. Hence this case cannot appear also. In particular, we have M2=b2=1, therefore there must be a prime closed geodesic c~α,l on (M~,F~) such that
(4.56)
i
(
c
~
α
,
l
)
=
2
,
k
0
(
c
~
α
,
l
)
=
1
for some α∈{1,2}. In fact, M2=1 implies there exist some α∈{1,2} and m∈ℕ such that C¯2(E,c~α,lm)=1. Thus by (4.2) and Proposition 2.1, we have i(c~α,lm)=2 and k0(c~α,lm)=1. Now if m=1, then (4.56) is true. If m>1, then
i
(
c
~
α
,
l
m
)
≥
i
(
c
~
α
,
l
m
-
1
)
+
ν
(
c
~
α
,
l
m
-
1
)
≥
⋯
≥
i
(
c
~
α
,
l
)
+
ν
(
c
~
α
,
l
)
>
2
by (4.2) and (4.3) provided ν(c~α,l)>0, this contradiction implies ν(c~α,l)=0, and then (4.56) holds by Proposition 2.1. Thus we have
C
¯
2
N
+
2
(
E
,
c
~
α
,
l
2
m
α
+
1
)
=
ℚ
,
by (3.11), (4.56) and Proposition 2.3. By Lemma 4.1, we have
i
(
c
~
1
,
l
2
m
1
)
+
ν
(
c
~
1
,
l
2
m
1
)
=
2
N
+
2
,
C
¯
2
N
+
2
(
E
,
c
~
1
,
l
2
m
1
)
≠
0
for 1≤l≤n1. Thus we have
(4.57)
2
=
b
2
N
+
2
=
M
2
N
+
2
≥
1
+
n
1
.
This implies that n1=1.
Step 3.
We have n2=1 provided c~1,l belongs to Cases 1.1 or 1.2 in Step 1.
In fact, by Cases 1.1 and 1.2, we can find (N′,m1′,m2′) such that
i
(
c
~
2
,
l
2
m
2
′
)
+
ν
(
c
~
2
,
l
2
m
2
′
)
=
2
N
′
+
2
,
C
¯
2
N
′
+
2
(
E
,
c
~
2
,
l
2
m
2
′
)
≠
0
for 1≤l≤n2. As above, C¯2N′+2(E,c~α,l2mα′+1)=ℚ. Then as in (4.57), we have 2=b2N′+2=M2N′+2≥1+n2. This implies n2=1.
Step 4.
We have n2=1 provided c~1,l belongs to Case 1.3 in Step 1.
Firstly we show
(4.58)
i
(
c
~
1
,
l
2
m
1
-
m
)
<
2
N
-
2
for all
m
≥
2
.
In fact, we have Pc~1,l=R(ϕ1)⋄R(ϕ2) with ϕiπ∉ℚ for i=1,2. By (4.3) and (4.6), if i(c~1,l)>2, then (4.58) holds. Thus it remains to consider the case i(c~1,l)=2. By [33, Lemma 2] and Theorem 3.5, we have
(4.59)
i
^
(
c
~
1
,
l
)
=
i
(
c
~
1
,
l
)
+
p
-
+
p
0
-
r
+
∑
i
=
1
r
ϕ
i
π
=
i
(
c
~
1
,
l
)
-
2
+
∑
i
=
1
2
ϕ
i
π
>
2
.
Plugging i(c~1,l)=2 into (4.59) yields
∑
i
=
1
2
(
ϕ
i
π
-
1
)
>
0
.
Thus we may assume without loss of generality that ϕ1∈(π,2π). By Theorem 3.5 we have
(4.60)
i
(
c
~
1
,
l
m
)
=
m
(
i
(
c
~
1
,
l
)
-
2
)
+
2
∑
i
=
1
2
E
(
m
ϕ
i
2
π
)
-
2
=
2
∑
i
=
1
2
E
(
m
ϕ
i
2
π
)
-
2
.
By (4.3) and (4.6), in order to prove (4.58), it suffices to prove
(4.61)
i
(
c
~
1
,
l
2
m
1
-
2
)
<
i
(
c
~
1
,
l
2
m
1
-
1
)
.
By (4.60), in order to prove (4.61), it is sufficient to prove
(4.62)
E
(
(
2
m
1
-
2
)
ϕ
1
2
π
)
<
E
(
(
2
m
1
-
1
)
ϕ
1
2
π
)
.
In order to satisfy (4.62), it is sufficient to choose
δ
<
min
{
ϕ
1
π
-
1
,
1
-
ϕ
1
2
π
}
,
where δ is given by (3.14). This proves (4.58). Thus by (4.11), (4.14), (4.52), (4.58) and Proposition 2.1, we have
C
¯
2
N
-
2
(
E
,
c
~
1
,
l
2
m
1
-
1
)
=
ℚ
,
C
¯
2
N
-
2
(
E
,
c
~
1
,
l
m
)
=
0
for all
m
≠
2
m
1
-
1
.
Thus we have
2
=
b
2
N
-
2
=
M
2
N
-
2
≥
1
+
n
2
.
This implies that n2=1.
Now we can complete the proof of Theorem 1.5 as follows: by Steps 2–4 and Proposition 2.1, we have
(4.63)
2
=
b
2
N
=
M
2
N
=
1
.
In fact, by the proof of Step 2, we have C¯q(E,c~j,l2mj)=ℚ for q=2N or 2N+2 by Proposition 2.2, thus (4.63) holds. This contradiction proves Theorem 1.5. ∎
