Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces

In this paper, let $n\geq2$ be an integer, $P=diag(-I_{n-\kappa},I_\kappa,-I_{n-\kappa},I_\kappa)$ for some integer $\kappa\in[0, n-1)$, and $\Sigma \subset {\bf R}^{2n}$ be a partially symmetric compact convex hypersurface, i.e., $x\in \Sigma$ implies $Px\in\Sigma$. We prove that if $\Sigma$ is $(r,R)$-pinched with $\frac{R}{r}<\sqrt{\frac{5}{3}}$, then $\Sigma$ carries at least two geometrically distinct P-symmetric closed characteristics which possess at least $2n-4\kappa$ Floquet multipliers on the unit circle of the complex plane.


Introduction and main results
In this paper, we consider the stability of P-symmetric closed characteristics on partially symmetric hypersurfaces in R 2n . Let Σ be a C 3 compact hypersurface in R 2n , 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 U contains the origin. We consider closed characteristics (τ, y) on Σ, which are solutions of the following problem ẏ(t) = JN Σ (y(t)), y(t) ∈ Σ, ∀ t ∈ R, y(τ ) = y(0), (1.1) where J =   0 −I n I n 0   , I n is the identity matrix in R n and N Σ (y) is the outward normal unit vector of Σ at y normalized by the condition N Σ (y) · y = 1. Here a · b denotes the standard inner product of a, b ∈ R 2n . 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 s i , t i ∈ R k i with i = 1, 2, we denote by (s 1 , t 1 ) ⋄ (s 2 , t 2 ) = (s 1 , s 2 , t 1 , t 2 ). Fixing an integer κ with 0 ≤ κ < n − 1, let P = diag(−I n−κ , I κ , −I n−κ , I κ ) and H κ (2n) = {Σ ∈ H(2n) | x ∈ Σ implies P x ∈ Σ}. For Σ ∈ H κ (2n), let Σ(κ) = {z ∈ R 2κ | 0 ⋄ z ∈ Σ}, where 0 is the origin in R 2n−2κ . As in [DoL1], A closed characteristic (τ, y) on Σ ∈ H κ (2n) is P-asymmetric if y(R) ∩ P y(R) = ∅, it is P-symmetric if y(R) = P y(R) with y = y 1 ⋄ y 2 and y 1 = 0, or it is P-fixed if y(R) = P y(R) and y = 0 ⋄ y 2 , where y 1 ∈ R 2(n−κ) , y 2 ∈ R 2κ . We call a closed characteristic (τ, y) is P-invariant if y(R) = P y(R). Then a P-invariant closed characteristic is P-symmetric or P-fixed.
(1.2) Denote by J (Σ, α) the set of all solutions (τ, y) of the 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.
As in Definition 5.1.6 of [Eke1], a C 3 hypersurface Σ bounding a compact convex set U , containing 0 in its interior is (r, R)-pinched, with 0 < r ≤ R, if: For the existence, multiplicity and stability of closed characteristics on convex compact hyper- Then there exist at least two geometrically distinct P-symmetric closed characteristics which possess at least 2n − 4κ Floquet multipliers on the unit circle of the complex plane.
Remark 1.2. In the above Theorem 1.1, let κ = 0, the P-symmetric closed characteristic is just symmetric and the P-fixed closed characteristics vanish, so Theorem 1.1 covers Theorem 1.1 of [Liu1].
In this paper, let N, N 0 , Z, Q, R and C denote the sets of natural integers, non-negative integers, integers, rational numbers, real numbers and complex numbers respectively. Denote by a · b and |a| the standard inner product and norm in R 2n . Denote by ·, · and · the standard L 2 -inner product and L 2 -norm. For an S 1 -space X, we denote by X S 1 the homotopy quotient of X by S 1 , i.e., X S 1 = S ∞ × S 1 X, where S ∞ is the unit sphere in an infinite dimensional complex Hilbert space. we define the functions Specially, φ(a) = 0 if a ∈ Z, and φ(a) = 1 if a / ∈ Z. We use Q coefficients for all homological modules.

A variational structure for P-invariant closed characteristics
In the rest of this paper, we fix a Σ ∈ H κ (2n). In this section, we review a variational structure for P-invariant closed characteristics established in [Liu2].
As in [Liu2], we associate with U a convex function H a . Consider the fixed period problem (2.1) Then by Proposition 2.2 of [Liu2], 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 2 κ 0, Define a linear operator Π κ : L 2 κ 0, 1 2 → L 2 κ 0, 1 2 by for any u = u 1 ⋄ u 2 ∈ L 2 κ 0, 1 2 . The corresponding Clarke-Ekeland dual action functional is defined by Proposition 2.6 of [Liu2], Ψ a is C 1,1 on L 2 κ 0, 1 2 and satisfies the Palais-Smale condition. Suppose x is a solution of (2.1). Then u =ẋ is a critical point of Ψ a . Conversely, suppose u is a critical point of Ψ a . Then there exists a unique ξ ∈ R 2n such that Π κ u − ξ is a solution of (2.1). In particular, solutions of (2.1) are in one to one correspondence with critical points of Ψ a . Moreover, Ψ a (u) < 0 for every critical point u = 0 of Ψ a .
Suppose u is a nonzero critical point of Ψ a . Then the formal Hessian of Ψ a at u is defined by which defines an orthogonal splitting L 2 κ 0, 1 2 = E − ⊕ E 0 ⊕ E + of L 2 κ 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) = dimE 0 . cf. Definition 2.10 of [Liu2].
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 As on Page 199 of [Eke1], we choose some α ∈ (1, 2) and associate with U a convex function . The corresponding Clarke-Ekeland dual action functional on L 2 κ 0, 1 2 is defined by where H * is the Fenchel transform of H.
For any ι ∈ R, we denote by As in Section 2 of [Liu2], we define whereÎ is the Fadell-Rabinowitz index defined above. Then By Propositions 2.15 and 2.16 of [Liu2], we have Proposition 2.1. Every c i is a critical value of Ψ. If c i = c j for some i < j, then there are infinitely many geometrically distinct P-invariant closed characteristics on Σ.
where u is a critical point of Ψ a corresponding to u α in the natural sense. In particular, we have

Index iteration theory for P-symmetric closed characteristics
In this section, we review the index iteration theory for P-symmetric closed characteristics which was studied in Section 3 of [Liu2].
Then we have u corresponds to P-symmetric closed characteristic (τ, y). Then we have Now we compute i(u 2m−1 ) via the index iteration method in [Lon1] and [DoL1]. First we recall briefly an index theory for symplectic paths. All the details can be found in [Lon1], [DoL1] and [Liu2].
In the following of this section, we assume P is some matrix of pattern As usual, the symplectic group Sp(2n) is defined by whose topology is induced from that of R 4n 2 . For τ > 0 we are interested in paths in Sp(2n): which is equipped with the topology induced from that of Sp(2n). The following real function was introduced in [DoL1]: 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 [DoL1]: For any M ∈ Sp(2n) 0 P,ω , we define a co-orientation of Sp(2n) 0 P,ω at M by the positive direction d dt M e tǫJ | t=0 of the path M e tǫJ with 0 ≤ t ≤ 1 and ǫ > 0 being sufficiently small. Let For any two continuous arcs ξ and η : [0, τ ] −→ Sp(2n) with ξ(τ ) = η(0), it is defined as usual: Given any two 2m k × 2m k matrices of square block form , the ⋄-product of M 1 and M 2 is defined by the following 2(m 1 + m 2 ) × 2(m 1 + m 2 ) matrix 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 γ j ∈ P τ (2n j ) with j = 0 and 1, let γ 1 ⋄γ 2 (t) = γ 1 (t)⋄γ 2 (t) for all t ∈ [0, τ ].
Let Ω 0 (M ) be the path connected component containing M = γ(τ ) of the set Here Ω 0 (M ) is called the homotopy component of M in Sp(2n).
In [Lon1], the following symplectic matrices were introduced as basic normal forms: Splitting numbers possess the following properties: Lemma 3.4.(cf. Proposition 3.8 of [DoL1]) Let (p ω (M P ), q ω (M P )) denote the Krein type of M P at ω. For any M ∈ Sp(2n) and ω ∈ U, the splitting numbers S ± M (P, ω) are well defined and satisfy the following properties.
(i) S ± M (P, ω) = S ± M P (ω), where the right-hand side is the splitting numbers given by Definition 9.1.4 of [Lon1].
We have the following where each M i is a basic normal form listed in (3.12)-(3.15) for 1 ≤ i ≤ l.

Proof of the main theorem
In this section, we give the proof of the main theorem.
Firstly, we point out a minor error in Example 6 on Page 278 of [DoL2] which is useful for us: Lemma 4.1. For any c > 0, we have Proof. Note that the definition of E(a) in our paper is different from that in [DoL2] by 1. In the proof of Example 6 of [DoL2], t = s k = (2kπ + 3 2 π)/c should be changed into t = s k = (2kπ + π)/c, and the expression in (3.14) of [DoL2] is wrong. One can easily verify our expression in (4.1) is right.
Note that since H 2 (·) is positive homogeneous of degree-two, by the (r, R)-pinched condition we have Comparing with the theorem of Morse-Schoenberg in the study of geodesics, we have the following Proposition 4.2. Let Σ ∈ H κ (2n) which is (r, R)-pinched. Suppose u 2m−1 is a nonzero critical point of Ψ a such that u corresponds to a P-symmetric closed characteristic (τ, y), and τ 2 = A(τ, y) as in Lemma 3.1. Then we have the following for some l ∈ N.
That is, since we prove our claim.
Now we give the proof of the main theorem.