Non-hyperbolic P-Invariant closed characteristics on partially symmetric compact convex hypersurfaces

Let n ≥ 2 be an integer, P = diag(-I_n-κ, I_κ, -I_n-κ, I_κ) for some integer κ ϵ [0,n], and let Σ ⊂ ℝ²ⁿ be a partially symmetric compact convex hypersurface, i.e., x ϵ Σ implies P_x ϵ Σ, and (r,R)-pinched. In this paper, we prove that when R/r √< 5/3 and 0 ≤ κ ≤ [n-1/2], there exist at least E(n-2κ-1/2 ) + E(n-2 κ-1/3) non-hyperbolic P-invariant closed characteristics on Σ. In addition, when R/r < √3/2, [n+1/2] ≤ κ ≤ n and Σ carries exactly n P-invariant closed characteristics, then there exist at least 2E (2κ-n-1/4) + (n-κ-1/3) non-hyperbolic P-invariant closed characteristics on Σ, where the function E(a) is defined as E(a) = min{k ϵℤ| k ≥a} for any a ϵ ℝ.