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

Abstract

Let n≥2{n\geq 2} be an integer, P=diag⁢(-In-κ,Iκ,-In-κ,Iκ){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⊂ℝ2⁢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-12]{0\leq\kappa\leq[\frac{n-1}{2}]}, there exist at least E⁢(n-2⁢κ-12)+E⁢(n-2⁢κ-13){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+12]≤κ≤n{[\frac{n+1}{2}]\leq\kappa\leq n} and Σ carries exactly nP-invariant closed characteristics, then there exist at least 2⁢E⁢(2⁢κ-n-14)+E⁢(n-κ-13){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}}.

Keywords

• compact convex hypersurfaces
• non-hyperbolic
• hamiltonian system
• 58e05
• 37j45
• 34c25