Non-hyperbolic P-Invariant Closed Characteristics on Partially Symmetric Compact Convex Hypersurfaces

Abstract

Abstract Let n ≥ 2 {n\geq 2} be an integer, P = diag ⁢ ( - I n - κ , I κ , - I n - κ , I κ ) {P=\mathrm{diag}(-I_{n-\kappa},I_{\kappa},-I_{n-\kappa},I_{\kappa})} for some integer κ ∈ [ 0 , n ] {\kappa\in[0,n]} , and let Σ ⊂ ℝ 2 ⁢ n {\Sigma\subset{\mathbb{R}}^{2n}} be a partially symmetric compact convex hypersurface, i.e., x ∈ Σ {x\in\Sigma} implies P ⁢ x ∈ Σ {Px\in\Sigma} , and ( r , R ) {(r,R)} -pinched. In this paper, we prove that when R / r < 5 / 3 {{R/r}<\sqrt{5/3}} and 0 ≤ κ ≤ [ n - 1 2 ] {0\leq\kappa\leq[\frac{n-1}{2}]} , there exist at least E ⁢ ( n - 2 ⁢ κ - 1 2 ) + E ⁢ ( n - 2 ⁢ κ - 1 3 ) {E(\frac{n-2\kappa-1}{2})+E(\frac{n-2\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ. In addition, when R / r < 3 / 2 {{R/r}<\sqrt{3/2}} , [ n + 1 2 ] ≤ κ ≤ n {[\frac{n+1}{2}]\leq\kappa\leq n} and Σ carries exactly n P-invariant closed characteristics, then there exist at least 2 ⁢ E ⁢ ( 2 ⁢ κ - n - 1 4 ) + E ⁢ ( n - κ - 1 3 ) {2E(\frac{2\kappa-n-1}{4})+E(\frac{n-\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ, where the function E ⁢ ( a ) {E(a)} is defined as E ⁢ ( a ) = min ⁡ { k ∈ ℤ ∣ k ≥ a } {E(a)=\min{\{k\in{\mathbb{Z}}\mid k\geq a\}}} for any a ∈ ℝ {a\in\mathbb{R}} .