1 Introduction and Main Results
In this paper, we consider the number of non-hyperbolic P-invariant closed characteristics on partially symmetric hypersurfaces in ℝ2n under a pinching condition. Let Σ be a C3 compact hypersurface in ℝ2n, bounding a strictly convex compact set U with non-empty interior, where n≥2. We denote the set of all such hypersurfaces by ℋ(2n). Without loss of generality, we suppose that U contains the origin. We consider closed characteristics (τ,y) on Σ, which are solutions of the following problem:
(1.1)
{
y
˙
(
t
)
=
J
N
Σ
(
y
(
t
)
)
,
y
(
t
)
∈
Σ
for all
t
∈
ℝ
,
y
(
τ
)
=
y
(
0
)
,
where
J
=
(
0
-
I
n
I
n
0
)
,
I
n
is the identity matrix in ℝn and NΣ(y) is the outward normal unit vector of Σ at y, normalized by the condition NΣ(y)⋅y=1. Here a⋅b denotes the standard inner product of a,b∈ℝ2n. A closed characteristic (τ,y) is prime if τ is the minimal period of y. Two closed characteristics (τ,x) and (σ,y) are geometrically distinct if x(ℝ)≠y(ℝ). We denote by 𝒥(Σ) the set of all closed characteristics (τ,y) on Σ with τ being the minimal period of y. For any si,ti∈ℝki, with i=1,2, we define (s1,t1)⋄(s2,t2)=(s1,s2,t1,t2). Fixing an integer κ, with 0≤κ≤n, let P=diag(-In-κ,Iκ,-In-κ,Iκ) and ℋκ(2n)={Σ∈ℋ(2n)∣x∈Σ⇒Px∈Σ}. For Σ∈ℋκ(2n), let Σ(κ)={z∈ℝ2κ∣0⋄z∈Σ}, where 0 is the origin in ℝ2n-2κ. As in [1], a closed characteristic (τ,y) on Σ∈ℋκ(2n) is P-asymmetric if y(ℝ)∩Py(ℝ)=∅, it is P-symmetric if y(ℝ)=Py(ℝ), with y=y1⋄y2 and y1≠0, and it is P-fixed if y(ℝ)=Py(ℝ) and y=0⋄y2, where y1∈ℝ2(n-κ), y2∈ℝ2κ. We say that a closed characteristic (τ,y) is P-invariant if y(ℝ)=Py(ℝ). Then a P-invariant closed characteristic is P-symmetric or P-fixed.
Let j:ℝ2n→ℝ be the gauge function of Σ, i.e., j(λx)=λ for x∈Σ and λ≥0. Then j∈C3(ℝ2n∖{0},ℝ)∩C0(ℝ2n,ℝ) and Σ=j-1(1). Fix a constant α∈(1,+∞) and define the Hamiltonian Hα:ℝ2n→[0,+∞) by
H
α
(
x
)
:=
j
(
x
)
α
.
Then Hα∈C3(ℝ2n∖{0},ℝ)∩C0(ℝ2n,ℝ) is convex and Σ=Hα-1(1). It is well known that problem (1.1) is equivalent to the following energy problem of the Hamiltonian system:
(1.2)
{
y
˙
(
t
)
=
J
H
α
′
(
y
(
t
)
)
,
H
α
(
y
(
t
)
)
=
1
for all
t
∈
ℝ
,
y
(
τ
)
=
y
(
0
)
.
Denote by 𝒥(Σ,α) the set of all solutions (τ,y) of problem (1.2), where τ is the minimal period of y. Note that elements in 𝒥(Σ) and 𝒥(Σ,α) are in one-to-one correspondence with each other. Let (τ,y)∈𝒥(Σ,α). We call the fundamental solution γy:[0,τ]→Sp(2n), with γy(0)=I2n, of the linearized Hamiltonian system
z
˙
(
t
)
=
J
H
α
′′
(
y
(
t
)
)
z
(
t
)
for all
t
∈
ℝ
the associated symplectic path of (τ,y). The eigenvalues of γy(τ) are called Floquet multipliers of (τ,y). By [3, Proposition 1.6.13], the Floquet multipliers with their multiplicities and Krein type numbers of (τ,y)∈𝒥(Σ,α) do not depend on the particular choice of the Hamiltonian function in (1.2). As in [15, Chapter 15], for any symplectic matrix M, we define the elliptic height e(M) of M by the total algebraic multiplicity of all eigenvalues of M on the unit circle 𝕌 in the complex plane ℂ. And for any (τ,y)∈𝒥(Σ,α), we define e(τ,y)=e(γy(τ)) and call (τ,y)elliptic or hyperbolic if e(τ,y)=2n or e(τ,y)=2, respectively.
As in [3, Definition 5.1.6], a C3 hypersurface Σ bounding a compact convex set U containing 0 in its interior is (r,R)-pinched, with 0<r≤R, if
|
y
|
2
R
-
2
≤
1
2
(
H
2
′′
(
x
)
y
,
y
)
≤
|
y
|
2
r
-
2
for all
x
∈
Σ
.
For the existence, multiplicity and stability of closed characteristics on convex compact hypersurfaces in ℝ2n we refer to [17, 23, 5, 6, 8, 4, 18, 10, 16, 22, 21] and the references therein. It is very interesting to consider closed characteristics on hypersurfaces with special symmetries. In [20, 12, 24], the multiplicity of closed characteristics with symmetries on convex compact hypersurfaces without pinching conditions was studied. For the stability problem of closed characteristics with symmetries, Hu and Sun [9] studied the index theory and stability of periodic solutions in Hamiltonian systems with symmetries. As an application they studied the stability of figure-eight orbit, due to Chenciner and Montgomery, in planar three-body problems with equal masses. In [11], Liu studied the ellipticity and non-hyperbolicity of symmetric closed characteristics on central symmetric compact convex hypersurfaces under a pinching condition. In [2], Dong and Long proved that there exists at least one P-invariant closed characteristic which possesses at least 2n-4κ Floquet multipliers on the unit circle of the complex plane. In [14], Liu and Zhang obtained two such closed characteristics under a pinching condition. In this paper, we continue to consider the non-hyperbolicity of P-invariant closed characteristic under some pinching conditions.
Theorem 1.1.
Assume Σ∈Hκ(2n) and 0<r≤|x|≤R for all x∈Σ. Then the following hold:
When
R
/
r
<
5
/
3
and
0
≤
κ
≤
[
n
-
1
2
]
, there exist at least
E
(
n
-
2
κ
-
1
2
)
+
E
(
n
-
2
κ
-
1
3
)
non-hyperbolic
P
-invariant closed characteristics on Σ.
When
R
/
r
<
3
/
2
, [n+12]≤κ≤n and Σ carries exactly nP-invariant closed characteristics, then there exist at least 2E(2κ-n-14)+E(n-κ-13) non-hyperbolic P-invariant closed characteristics on Σ.
Remark 1.2.
In Theorem 1.1, let κ=0. The P-symmetric closed characteristic is just central symmetric, the P-fixed closed characteristics vanish, and we have E(n-2κ-12)+E(n-2κ-13)=E(n-12)+E(n-13) for κ=0. Thus, Theorem 1.1 generalizes [11, Theorem 1.2].
In this paper, let ℕ,ℤ,ℚ,ℝ and ℂ denote the sets of natural integers, integers, rational numbers, real numbers and complex numbers, respectively. Denote by a⋅b and |a| the standard inner product and norm in ℝ2n. We use ℚ coefficients for all homological modules. We define the functions [a]=max{k∈ℤ∣k≤a} and E(a)=min{k∈ℤ∣k≥a}.
2 A Variational Structure for P-Invariant Closed Characteristics
In the rest of this paper, we fix a Σ∈ℋκ(2n). In this section, we review the variational structure for P-invariant closed characteristics established in [12].
As in [12], we associate with U a convex function Ha. Consider the fixed period problem
(2.1)
{
x
˙
(
t
)
=
J
H
a
′
(
x
(
t
)
)
,
x
(
1
/
2
)
=
P
x
(
0
)
.
Then, by [12, Proposition 2.2], the nonzero solutions of (2.1) are in one-to-one correspondence with P-symmetric closed characteristics with period τ<a and P-fixed closed characteristics with period τ/2<a/2. Let
L
κ
2
(
0
,
1
/
2
)
=
{
u
=
u
1
⋄
u
2
∈
L
2
(
(
0
,
1
/
2
)
,
ℝ
2
n
)
|
u
1
∈
L
2
(
(
0
,
1
/
2
)
,
ℝ
2
n
-
2
κ
)
,
u
2
∈
L
2
(
(
0
,
1
/
2
)
,
ℝ
2
κ
)
,
u
(
1
/
2
)
=
P
u
(
0
)
,
∫
0
1
/
2
u
2
(
t
)
d
t
=
0
}
.
Define a linear operator Πκ:Lκ2(0,1/2)→Lκ2(0,1/2) by
(
Π
κ
u
)
(
t
)
=
x
1
(
t
)
⋄
x
2
(
t
)
,
where
x
1
(
t
)
=
∫
0
t
u
1
(
τ
)
𝑑
τ
-
1
2
∫
0
1
/
2
u
1
(
τ
)
𝑑
τ
,
x
2
(
t
)
=
∫
0
t
u
2
(
τ
)
𝑑
τ
-
2
∫
0
1
/
2
𝑑
t
∫
0
t
u
2
(
τ
)
𝑑
τ
for any u=u1⋄u2∈Lκ2(0,1/2).
The corresponding Clarke–Ekeland dual action functional is defined by
Ψ
a
(
u
)
=
∫
0
1
/
2
(
1
2
J
u
⋅
Π
κ
u
+
G
a
(
-
J
u
)
)
𝑑
t
,
where Ga is the Fenchel transform of Ha defined by Ga(x)=sup{x⋅y-Ha(y)∣y∈ℝ2n}. By [12, Proposition 2.6], Ψa is C1,1 on Lκ2(0,1/2) and satisfies the Palais–Smale condition. Suppose x is a solution of (2.1). Then u=x˙ is a critical point of Ψa. Conversely, suppose u is a critical point of Ψa. Then there exists a unique ξ∈ℝ2n such that Πκu-ξ is a solution of (2.1). In particular, the solutions of (2.1) are in one-to-one correspondence with the critical points of Ψa. Moreover, Ψa(u)<0 for every critical point u≠0 of Ψa.
Suppose u is a nonzero critical point of Ψa. Then the formal Hessian of Ψa at u is defined by
Q
a
(
v
,
v
)
=
∫
0
1
/
2
(
J
v
⋅
Π
κ
v
+
G
a
′′
(
-
J
u
)
J
v
⋅
J
v
)
𝑑
t
,
which defines an orthogonal splitting Lκ2(0,1/2)=E-⊕E0⊕E+ of Lκ2(0,1/2) into negative, zero and positive subspaces. The index of u is defined by i(u)=dimE- and the nullity of u is defined by ν(u)=dimE0, cf. [12, Definition 2.10].
Note that we can identify Lκ2(0,1/2) with the space
{
u
∈
L
2
(
ℝ
/
ℤ
,
ℝ
2
n
)
∣
u
|
(
0
,
1
/
2
)
∈
L
κ
2
(
0
,
1
/
2
)
,
u
(
t
+
1
/
2
)
=
P
u
(
t
)
}
.
Then we have a natural S1-action on Lκ2(0,1/2), defined by θ*u(t)=u(θ+t) for all θ∈S1≡ℝ/ℤ and t∈ℝ. By [12, Lemma 2.8], Ψa is S1-invariant. Hence, if u is a critical point of Ψa, then the whole orbit S1⋅u is formed by critical points of Ψa. Denote by crit(Ψa) the set of critical points of Ψa. Then crit(Ψa) is compact by the Palais–Smale condition.
Recall that for a principal U(1)-bundle E→B, the Fadell–Rabinowitz index (cf. [7]) of E is defined to be sup{k∣c1(E)k-1≠0}, where c1(E)∈H2(B,ℚ) is the first rational Chern class. For a U(1)-space, i.e., a topological space X with a U(1)-action, the Fadell–Rabinowitz index is defined to be the index of the bundle X×S∞→X×U(1)S∞, where S∞→CP∞ is the universal U(1)-bundle.
As in [3, p. 199], we choose some α∈(1,2) and associate with U a convex function H(x)=j(x)α for all x∈ℝ2n. Consider the fixed period problem
{
x
˙
(
t
)
=
J
H
′
(
x
(
t
)
)
,
x
(
1
/
2
)
=
P
x
(
0
)
.
The corresponding Clarke–Ekeland dual action functional on Lκ2(0,1/2) is defined by
Ψ
(
u
)
=
∫
0
1
/
2
(
1
2
J
u
⋅
Π
κ
u
+
H
*
(
-
J
u
)
)
𝑑
t
for all
u
∈
L
κ
2
(
0
,
1
/
2
)
,
where H* is the Fenchel transform of H.
For any ι∈ℝ, we define
Ψ
ι
-
=
{
w
∈
L
κ
2
(
0
,
1
/
2
)
∣
Ψ
(
w
)
<
ι
}
.
As in [12, Section 2], we define
c
i
=
inf
{
δ
∈
ℝ
∣
I
^
(
Ψ
δ
-
)
≥
i
}
.
where I^ is the Fadell–Rabinowitz index defined above. Then
c
1
≤
c
2
≤
⋯
c
i
≤
c
i
+
1
≤
⋯
<
0
.
By [12, Propositions 2.15 and 2.16], we have the following propositions.
Proposition 2.1.
Every ci is a critical value of Ψ. If ci=cj for some i<j, then there are infinitely many geometrically distinct P-invariant closed characteristics on Σ.
Proposition 2.2.
For every i∈N, there exists a critical point uα of Ψ, found in Proposition 2.1, such that
Ψ
(
u
α
)
=
c
i
,
C
S
1
,
2
i
-
2
(
Ψ
a
,
S
1
⋅
u
)
≠
0
,
where u is a critical point of Ψa corresponding to uα in the natural sense. In particular, we have i(u)≤2(i-1)≤i(u)+ν(u)-1.
3 Index Theory for P-Invariant Closed Characteristics
In this section, we review the index theory for P-invariant closed characteristics which was studied in [12, Section 3].
Note that if (τ,y) is P-symmetric, then ((2m-1)τ,y) is P-symmetric for any m∈ℕ. Thus, ((2m-1)τ,y) corresponds to a critical point of Ψa via [12, Propositions 2.2 and 2.6]; we denote it by u2m-1. If (τ2,y) is P-fixed, then (mτ/2,y) is P-fixed for any m∈ℕ. Thus, (mτ/2,y) corresponds to a critical point of Ψa; we denote it by um. Recall that the action of a closed characteristic (τ,y) is defined by (cf. [3, p. 190])
A
(
τ
,
y
)
=
1
2
∫
0
τ
(
J
y
⋅
y
˙
)
𝑑
t
.
Now we consider the linear Hamiltonian system
(3.1)
{
ξ
˙
(
t
)
=
J
A
(
t
)
ξ
,
A
(
t
+
τ
2
/
2
)
=
P
A
(
t
)
P
,
where A(t)=H2′′(y(t)) and τ2=A(τ,y). Denote by iPE(A,k) and νPE(A,k) the index and nullity of the k-th iteration of system (3.1) defined by Dong and Long (cf. [2, Definition 3.4]), respectively. Denote by iP,1(γAk,P) and νP,1(γAk,P) the P-index and P-nullity of the k-th iteration of system (3.1) defined by Dong and Long (cf. [1, Section 3]), respectively, where γA is the fundamental solution of (3.1), with γA(0)=I2n. Then we have the following theorem.
Theorem 3.1 (cf. [12, Theorem 3.2]).
If
u
m
is a nonzero critical point of
Ψ
a
such that
u
corresponds to the
P
-fixed closed characteristic
(
τ
/
2
,
y
)
, then we have
i
(
u
m
)
=
i
P
,
1
(
γ
A
m
,
P
)
-
κ
,
ν
(
u
m
)
=
ν
P
,
1
(
γ
A
m
,
P
)
-
1
.
If
u
2
m
-
1
is a nonzero critical point of
Ψ
a
such that
u
corresponds to the
P
-symmetric closed characteristic
(
τ
,
y
)
, then we have
i
(
u
2
m
-
1
)
=
i
P
,
1
(
γ
A
2
m
-
1
,
P
)
-
κ
,
ν
(
u
2
m
-
1
)
=
ν
P
,
1
(
γ
A
2
m
-
1
,
P
)
-
1
.
Now we recall briefly the P-index theory for symplectic paths. All the details can be found in [15, 1, 12].
In this section, we assume that P is some matrix of pattern (-I2s-2t)⋄I2t, where 0≤t≤s.
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 the following paths in Sp(2n):
𝒫
τ
(
2
n
)
=
{
γ
∈
C
(
[
0
,
τ
]
,
Sp
(
2
n
)
)
∣
γ
(
0
)
=
I
2
n
}
,
which are equipped with the topology induced from that of Sp(2n). The following real function was introduced in [1]:
D
P
,
ω
(
M
)
=
(
-
1
)
n
-
1
ω
¯
n
det
(
M
-
ω
P
)
for all
ω
∈
𝕌
,
M
∈
Sp
(
2
n
)
,
where 𝕌 is the unit circle in the complex plane. Thus, for any ω∈𝕌, the following codimension 1 hypersurface in Sp(2n) is defined in [1]:
Sp
(
2
n
)
P
,
ω
0
=
{
M
∈
Sp
(
2
n
)
∣
D
P
,
ω
(
M
)
=
0
}
.
For any M∈Sp(2n)P,ω0, we define a co-orientation of Sp(2n)P,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
)
P
,
ω
*
=
Sp
(
2
n
)
∖
Sp
(
2
n
)
P
,
ω
0
,
𝒫
P
,
τ
,
ω
*
(
2
n
)
=
{
γ
∈
𝒫
τ
(
2
n
)
∣
γ
τ
∈
Sp
(
2
n
)
P
,
ω
*
}
,
𝒫
P
,
τ
,
ω
0
(
2
n
)
=
𝒫
τ
(
2
n
)
∖
𝒫
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
-
1
)
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 [15], 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=0,1, let γ1⋄γ2(t)=γ1(t)⋄γ2(t) for all t∈[0,τ].
A special path ξn∈𝒫τ(2n) is defined by
ξ
n
(
t
)
=
(
2
-
t
/
τ
0
0
(
2
-
t
/
τ
)
-
1
)
⋄
n
for
0
≤
t
≤
τ
.
Definition 3.2 (cf. [1] and [12, Definition 3.3]).
For any ω∈𝕌 and M∈Sp(2n), via [15, Definition 5.4.4], we define
ν
P
,
ω
(
M
)
=
dim
ℂ
ker
ℂ
(
M
-
ω
P
)
.
For any τ>0 and γ∈𝒫τ(2n), define
ν
P
,
ω
(
γ
)
=
ν
ω
(
γ
P
)
=
ν
P
,
ω
(
γ
(
τ
)
)
.
If γ∈𝒫P,τ,ω*(2n), define
(3.2)
i
P
,
ω
(
γ
)
=
[
Sp
(
2
n
)
P
,
ω
0
:
γ
∗
ξ
n
]
,
where the right-hand side of (3.2) 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∈𝒫P,τ,ω0(2n), we let ℱ(γ) be the set of all open neighborhoods of γ in 𝒫τ(2n), and define
i
P
,
ω
(
γ
)
=
sup
U
∈
ℱ
(
γ
)
inf
{
i
P
,
ω
(
β
)
∣
β
∈
U
∩
𝒫
P
,
τ
,
ω
*
(
2
n
)
}
.
Then (iP,ω(γ),νP,ω(γ))∈ℤ×{0,1,…,2n} is called the P-index function of γ at ω.
For any M∈Sp(2n) and ω∈𝕌, the splitting numbersSM±(P,ω) of M at (P,ω) are defined by
S
M
±
(
P
,
ω
)
=
lim
ϵ
→
0
+
i
P
,
ω
exp
(
±
-
1
ϵ
)
(
γ
)
-
i
P
,
ω
(
γ
)
for any path γ∈𝒫τ(2n) satisfying γ(τ)=M.
Let Ω0(M) be the path connected component 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 [15], 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.
Splitting numbers possess the following properties.
Lemma 3.3 (cf. [1, Proposition 3.8]).
Let (pω(MP),qω(MP)) denote the Krein type of MP at ω. For any M∈Sp(2n) and ω∈U, the splitting numbers SM±(P,ω) are well defined and satisfy the following properties:
S
M
±
(
P
,
ω
)
=
S
M
P
±
(
ω
)
, where the right-hand side is the splitting numbers given by
[
15
, Definition 9.1.4].
S
M
+
(
P
,
ω
)
-
S
M
-
(
P
,
ω
)
=
p
ω
(
M
P
)
-
q
ω
(
M
P
)
.
S
M
±
(
P
,
ω
)
=
S
N
±
(
P
,
ω
)
if
N
P
∈
Ω
0
(
M
P
)
.
S
M
1
⋄
M
2
±
(
P
,
ω
)
=
S
M
1
±
(
P
1
,
ω
)
+
S
M
2
±
(
P
2
,
ω
)
for
M
j
,
P
j
∈
Sp
(
2
n
j
)
, with
n
j
∈
{
1
,
…
,
n
}
, satisfying
P
=
P
1
⋄
P
2
and
n
1
+
n
2
=
n
.
S
M
±
(
P
,
ω
)
=
0
if
ω
∉
σ
(
M
P
)
.
By [1, Proposition 3.10], we have the following Bott-type formula for the (P,ω)-index.
Lemma 3.4.
For any γ∈Pτ(2n), z∈U and m∈N, we have
i
P
m
,
z
(
γ
m
,
P
)
=
∑
ω
m
=
z
i
P
,
ω
(
γ
)
,
ν
P
m
,
z
(
γ
m
,
P
)
=
∑
ω
m
=
z
ν
P
,
ω
(
γ
)
.
4 Proof of the Main Theorem
In this section, we give the proof of the main theorem.
Lemma 4.1.
Let Σ∈Hκ(2n) be (r,R)-pinched. Suppose um is a nonzero critical point of Ψa such that u corresponds to a P-symmetric closed characteristic (τ,y) for m∈2N-1, or a P-fixed closed characteristic (τ/2,y) for m∈N and τ2=A(τ,y). Then we have
(4.1)
i
(
u
m
)
≥
2
n
l
,
if
m
τ
2
/
2
>
l
π
R
2
,
(4.2)
i
(
u
m
)
+
ν
(
u
m
)
≤
2
n
(
l
-
1
)
-
1
if
m
τ
2
/
2
<
(
l
-
1
/
2
)
π
r
2
,
for some l∈N.
Proof.
If u corresponds to a P-symmetric closed characteristic (τ,y) and m∈2ℕ-1, our lemma is just [14, Proposition 3]. If u corresponds to a P-fixed closed characteristic (τ/2,y) for m∈ℕ, our proof is completely the same as that of [14, Proposition 3]. ∎
By [13, Theorems 1.1 and 1.2], we have the following lemma.
Lemma 4.2.
Assume Σ∈Hκ(2n) and 0<r≤|x|≤R for all x∈Σ with R/r<2. Then the following hold:
There exist at least
n
-
κ
geometrically distinct
P
-symmetric closed characteristics
(
τ
i
,
y
i
)
on Σ for 1≤i≤n-κ, where τi is the minimal period of yi, and the actions A(τi,yi) satisfy
(4.3)
π
r
2
≤
A
(
τ
i
,
y
i
)
≤
π
R
2
.
There exist at least κ geometrically distinct P-fixed closed characteristics (σi,zi) on Σ for 1≤i≤κ, where σi is the minimal period of zi, and the actions A(σi,zi) satisfy (4.3) for 1≤i≤κ.
Lemma 4.3.
Let {(τ1,y1),…,(τn-κ,yn-κ)} be the P-symmetric closed characteristics found in Lemma 4.2. Then we have
(4.4)
Ψ
a
′
(
u
i
)
=
0
,
i
(
u
i
)
≤
2
(
i
-
1
)
≤
i
(
u
i
)
+
ν
(
u
i
)
-
1
,
for 1≤i≤n-κ, where ui is the unique critical point of Ψa corresponding to (τi,yi). Furthermore, if R/r<3/2, then for n-κ+1≤i≤n, (4.4) also holds for a critical point ui of Ψa corresponding to a prime closed characteristic (τi,yi) which is P-symmetric or P-fixed.
Proof.
By the proof of Lemma 4.2 and Proposition 2.2, (4.4) holds for 1≤i≤n-κ. For n-κ+1≤i≤n, by Proposition 2.2, we can find a closed characteristic (τi,yi) which is P-symmetric or P-fixed and corresponds to a critical point ui of Ψa satisfying (4.4). In the following, we prove that (τi,yi) is prime for n-κ+1≤i≤n when R/r<3/2.
Note that, by [3, Theorem V.1.4], for any closed characteristic (τ,y), we have A(τ,y)≥πr2. Then, by the pinching condition R/r<3/2, we obtain A(mτ,ym)≥mπr2>πR2 and A((2m-1)τ,y2m-1)/2≥(2m-1)πr2/2>πR2 for m≥2, which together with (4.1) give i(um)≥2n, where u is the unique critical point of Ψa corresponding to (τ,y). On the other hand, by (4.4), we have i(ui)≤2(i-1)≤2n-2 for i≤n, thus we must have that (τi,yi) is prime. ∎
Corollary 4.4.
Assume Σ∈Hκ(2n) and 0<r≤|x|≤R for all x∈Σ, with R/r<3/2. If Σ carries exactly nP-invariant closed characteristics, which we denote, as in Lemma 4.3, by {(τ1,y1),…,(τn,yn)}, then for 1≤i≤n-κ, (τi,yi) is prime P-symmetric, and for n-κ+1≤i≤n, (τi,yi) is prime P-fixed. In addition, for all 1≤i≤n, (4.3) holds.
Proof.
By Lemma 4.3, for 1≤i≤n-κ, (τi,yi) is prime P-symmetric, and for n-κ+1≤i≤n, (τi,yi) is prime P-fixed or P-symmetric. Since we have exactly nP-invariant closed characteristics by assumption, then, by Lemma 4.2 (ii), we know that for n-κ+1≤i≤n, (τi,yi) is prime P-fixed and they are just the P-fixed ones {(σi,zi)∣1≤i≤κ}. Thus, for all 1≤i≤n, (4.3) holds from Lemma 4.2. ∎
By [12, Proposition 2.13], we have that the fundamental solution γ=γy:[0,τ/2]→Sp(2n), with γy(0)=I2n, of the linearized Hamiltonian system
γ
y
˙
(
t
)
=
J
H
2
′′
(
y
(
t
)
)
γ
y
(
t
)
for all
t
∈
ℝ
satisfies
(4.5)
γ
y
(
τ
/
2
)
P
=
Q
y
-
1
(
I
2
⋄
M
)
Q
y
for some symplectic matrices Qy and M (cf. also [15, Lemmas 15.2.3 and 15.2.4]). We call γ=γy:[0,τ/2]→Sp(2n) the associated symplectic path of (τ,y).
Lemma 4.5.
Let (τ,y) (resp. (τ/2,y)) be a hyperbolic P-symmetric (resp. P-fixed) closed characteristic and let γ≡γy:[0,τ/2]→Sp(2n) be its associated symplectic path. Then the following hold:
i
P
,
-
1
(
γ
)
=
i
P
,
1
(
γ
)
+
1
,
i
P
,
1
(
γ
)
≥
max
(
κ
,
[
n
2
]
)
and
i
P
,
-
1
(
γ
)
≥
max
(
n
-
κ
,
[
n
2
]
+
1
,
κ
+
1
)
,
i
P
m
,
1
(
γ
m
,
P
)
=
m
(
i
P
,
1
(
γ
)
+
1
)
-
1
, νPm,1(γm,P)=2 for all m∈ℕ.
Proof.
Since (τ,y) or (τ/2,y) is hyperbolic, e(τ,y)=2. Noticing that e(τ,y)=e(γ(τ))=e((γ(τ/2)P)2), we have σ(M)∩𝕌=∅ in (4.5). Then (i) and (iii) follow from the definition of splitting numbers, [15, List 9.1.12] and Lemmas 3.3–3.4. By [1, Theorem 4.1], we have κ≤iP,1(γ) and n-κ≤iP,-1(γ), which, together with (i), gives (ii). ∎
Now we give the proof of the main theorem.
Let (τi,yi) be a hyperbolic P-invariant closed characteristic for some i∈[1,n]. By [12, Proposition 2.13], we have γyi(τi/2)P=Qyi-1(I2⋄Mi)Qyi, where σ(Mi)∩𝕌=∅, and Qyi,Mi are symplectic matrices. We first consider two cases.
Case 1.
(
τ
i
,
y
i
)
is a hyperbolic P-symmetric closed characteristic and R/r<5/3.
By Theorem 3.1 (ii) and Lemma 4.5 (iii), we have
i
(
u
i
3
)
+
ν
(
u
i
3
)
=
i
P
,
1
(
γ
i
3
)
-
κ
+
ν
P
,
1
(
γ
i
3
)
-
1
=
3
i
P
,
1
(
γ
i
)
+
2
-
κ
+
1
(4.6)
=
3
i
P
,
1
(
γ
i
)
-
κ
+
3
.
By Lemma 4.3, we have that (4.3) holds for 1≤i≤n-κ, which implies 3A(τi,yi)/2<(3-1/2)πr2. This, together with (4.2), implies
(4.7)
i
(
u
i
3
)
+
ν
(
u
i
3
)
≤
4
n
-
1
.
Combining (4.6) with (4.7), we obtain
(4.8)
i
P
,
1
(
γ
i
)
≤
[
4
n
+
κ
-
4
3
]
.
By Theorem 3.1 (ii) and (4.4), we obtain
(4.9)
i
P
,
1
(
γ
i
)
-
κ
=
i
(
u
i
)
=
2
(
i
-
1
)
.
Now we proceed our proof according to the value of κ. Subcase 1.1.0≤κ≤[n-12].
From (4.8), (4.9) and Lemma 4.5 (ii), we have
[
n
2
]
≤
i
P
,
1
(
γ
i
)
=
2
(
i
-
1
)
+
κ
≤
[
4
n
+
κ
-
4
3
]
,
which implies
(4.10)
E
(
n
-
2
κ
+
3
4
)
≤
i
≤
[
2
n
-
κ
+
1
3
]
.
On the other hand, note that iP,-1(γi)=iP,1(γi)+1, by Lemma 4.5 (i). Using this, together with (4.9) and Lemma 4.5 (ii), we get
n
-
κ
≤
i
P
,
-
1
(
γ
i
)
=
2
(
i
-
1
)
+
κ
+
1
.
This, together with (4.10), yields
(4.11)
E
(
n
-
2
κ
+
1
2
)
≤
i
≤
[
2
n
-
κ
+
1
3
]
.
Comparing (4.10) with (4.11) and noticing that 0≤κ≤[n-12], we obtain
(4.12)
E
(
n
-
2
κ
+
1
2
)
≤
i
≤
[
2
n
-
κ
+
1
3
]
.
Subcase 1.2.
[
n
+
1
2
]
≤
κ
≤
n
.
From (4.8), (4.9) and Lemma 4.5 (ii), we have
κ
≤
i
P
,
1
(
γ
i
)
=
2
(
i
-
1
)
+
κ
≤
[
4
n
+
κ
-
4
3
]
,
which implies
(4.13)
1
≤
i
≤
[
2
n
-
κ
+
1
3
]
.
On the other hand, note that iP,-1(γi)=iP,1(γi)+1, by Lemma 4.5 (i). Using this, together with (4.8), (4.9) and Lemma 4.5 (ii), we have
(4.14)
[
n
2
]
+
1
≤
i
P
,
-
1
(
γ
i
)
=
2
(
i
-
1
)
+
κ
+
1
≤
[
4
n
+
κ
-
4
3
]
+
1
.
Comparing (4.13) with (4.14) and noticing that [n+12]≤κ≤n, we obtain
1
≤
i
≤
[
2
n
-
κ
+
1
3
]
.
In this subcase, if we impose the condition R/r<3/2, then by Lemma 4.3 we have that (4.3) holds for 1≤i≤n-κ, which implies 3A(τi,yi)/2≥3πr2/2>πR2. This, together with (4.1), implies
(4.15)
i
(
u
i
3
)
≥
2
n
.
By Theorem 3.1 (ii) and Lemma 4.5 (iii), we have
(4.16)
i
(
u
i
3
)
=
i
P
,
1
(
γ
i
3
)
-
κ
=
3
i
P
,
1
(
γ
i
)
+
2
-
κ
.
Combining (4.9) with (4.15)–(4.16), we get
(4.17)
i
≥
E
(
n
-
κ
+
2
3
)
.
Case 2.
(
τ
i
,
y
i
)
is a hyperbolic P-fixed closed characteristic and R/r<3/2.
By Lemma 4.5 (iii), we have
(4.18)
i
(
γ
i
2
)
=
i
P
2
,
1
(
γ
i
2
,
P
)
=
2
i
P
,
1
(
γ
i
)
+
1
,
(4.19)
i
(
γ
i
2
)
+
ν
(
γ
i
2
)
-
1
=
i
P
2
,
1
(
γ
i
2
,
P
)
+
ν
P
2
,
1
(
γ
i
2
,
P
)
-
1
=
2
i
P
,
1
(
γ
i
)
+
2
.
On the other hand, by the pinching condition R/r<3/2, we have A(2τi,yi2)≤2πR2<3πr2, which, together with [19, Lemma 3.1] and [15, Theorem 15.1.1], gives
(4.20)
i
(
γ
i
2
)
≥
3
n
,
(4.21)
i
(
γ
i
2
)
+
ν
(
γ
i
2
)
-
1
≤
5
n
-
1
.
By (4.18)–(4.21), we obtain
(4.22)
E
(
3
n
-
1
2
)
≤
i
P
,
1
(
γ
i
)
≤
[
5
n
-
3
2
]
.
Since (τi,yi) is hyperbolic, by Theorem 3.1 (i) and (4.4), we have
(4.23)
i
P
,
1
(
γ
i
)
-
κ
=
i
(
u
i
)
=
2
(
i
-
1
)
.
In the following, we proceed our proof according to the value of κ. Subcase 2.1.0≤κ≤[n-12].
By (4.22) and (4.23), we have
E
(
3
n
-
1
2
)
≤
i
P
,
1
(
γ
i
)
=
2
(
i
-
1
)
+
κ
≤
[
5
n
-
3
2
]
.
This implies
(4.24)
E
(
3
n
-
2
κ
+
3
4
)
≤
i
≤
[
5
n
-
2
κ
+
1
4
]
,
which, together with 0≤κ≤[n-12] and i≤n, gives
(4.25)
E
(
3
n
-
2
κ
+
3
4
)
≤
i
≤
n
.
Subcase 2.2.
[
n
+
1
2
]
≤
κ
≤
n
.
In this subcase, (4.24) also holds. Since [5n-2κ+14]≤n and E(3n-2κ+34)≥1 for [n+12]≤κ≤n, we have
(4.26)
E
(
3
n
-
2
κ
+
3
4
)
≤
i
≤
[
5
n
-
2
κ
+
1
4
]
.
Now we can prove Theorem 1.1.
Proof of Theorem 1.1.
(i) We have the condition 0≤κ≤[n-12] and Σ is (r,R)-pinched with R/r<5/3.
Since 2>5/3, by Lemma 4.3, we obtain n-κ geometrically distinct P-symmetric closed characteristics {(τi,yi)∣1≤i≤n-κ} on Σ. From (4.12), there exist at most [2n-κ+13]-E(n-2κ+12)+1 hyperbolic ones in {(τi,yi)∣1≤i≤n-κ}, hence there exist at least
(4.27)
n
-
κ
-
[
2
n
-
κ
+
1
3
]
+
E
(
n
-
2
κ
+
1
2
)
-
1
=
E
(
n
-
2
κ
-
1
2
)
+
E
(
n
-
2
κ
-
1
3
)
non-hyperbolic P-symmetric closed characteristics on Σ. Since E(3n-2κ+34)≤n-κ+1, by (4.25), all of the closed characteristics {(τi,yi)∣n-κ+1≤i≤n} may be hyperbolic. In summary, we obtain at least E(n-2κ-12)+E(n-2κ-13) non-hyperbolic P-invariant closed characteristics on Σ.
(ii) We have the condition [n+12]≤κ≤n and Σ is (r,R)-pinched with R/r<3/2, which carries exactly nP-invariant closed characteristics.
By Corollary 4.4, we know that {(τi,yi)∣n-κ+1≤i≤n} are P-fixed. Since E(3n-2κ+34)≥n-κ+1, by (4.26), there exist at most [5n-2κ+14]-E(3n-2κ+34)+1 hyperbolic ones in {(τi,yi)∣n-κ+1≤i≤n}. Then there exist at least
(4.28)
κ
-
[
5
n
-
2
κ
+
1
4
]
+
E
(
3
n
-
2
κ
+
3
4
)
-
1
=
2
E
(
2
κ
-
n
-
1
4
)
non-hyperbolic P-fixed closed characteristics on Σ. By (4.17), at least E(n-κ+23)-1=E(n-κ-13) of the P-symmetric closed characteristics {(τi,yi)∣1≤i≤n-κ} are non-hyperbolic. In summary, we obtain at least 2E(2κ-n-14)+E(n-κ-13) non-hyperbolic P-invariant closed characteristics on Σ. The proof is complete. ∎
Remark 4.6.
We prove identities (4.27) and (4.28). We have
n
-
κ
-
[
2
n
-
κ
+
1
3
]
+
E
(
n
-
2
κ
+
1
2
)
-
1
=
-
(
[
2
n
-
κ
+
1
3
]
-
n
+
κ
)
+
E
(
n
-
2
κ
-
1
2
)
=
-
[
-
n
+
2
κ
+
1
3
]
+
E
(
n
-
2
κ
-
1
2
)
=
E
(
n
-
2
κ
-
1
3
)
+
E
(
n
-
2
κ
-
1
2
)
and
κ
-
[
5
n
-
2
κ
+
1
4
]
+
E
(
3
n
-
2
κ
+
3
4
)
-
1
=
-
[
n
-
2
κ
+
1
4
]
+
(
E
(
3
n
-
2
κ
+
3
4
)
-
1
-
n
+
κ
)
=
-
[
n
-
2
κ
+
1
4
]
+
E
(
-
n
+
2
κ
-
1
4
)
=
2
E
(
2
κ
-
n
-
1
4
)
.
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