Elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces in ${\mathbf{R}}^{2n}$

Abstract

Let $\Sigma$ be a compact convex hypersurface in ${\mathbf{R}}^{2n}$ which is P-cyclic symmetric, i.e., $x\in \Sigma$ implies $Px\in \Sigma$ with P being a $2n\times 2n$ symplectic orthogonal matrix and $P^k=I_{2n}$, where $n,k\ge 2$, $ker(P-I_{2n})=0$. In this paper, we first generalize Ekeland index theory for periodic solutions of convex Hamiltonian system to a index theory with P boundary value condition and study its relationship with Maslov P-index theory, then we use index theory to prove the existence of elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces in ${\mathbf{R}}^{2n}$ for a broad class of symplectic orthogonal matrix P.

24 Pages. arXiv admin note: text overlap with arXiv:0812.0041 by other authors