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

TL;DR

This paper extends index theory for convex Hamiltonian systems with P boundary conditions and proves the existence of elliptic and non-hyperbolic closed characteristics on symmetric hypersurfaces in symplectic space.

Contribution

It generalizes Ekeland index theory to P boundary conditions and establishes new existence results for closed characteristics on symmetric hypersurfaces.

Findings

Existence of elliptic closed characteristics on symmetric hypersurfaces.

Existence of non-hyperbolic closed characteristics.

Relationship between index theory and Maslov P-index.

Abstract

Let $\Sigma$ be a compact convex hypersurface in ${\bf R}^{2n}$ which is P-cyclic symmetric, i.e., $x\in \Sigma$ implies $Px\in\Sigma$ with P being a $2n\times2n$ symplectic orthogonal matrix and $P^k=I_{2n}$, where $n, k\geq2$, $ker(P-I_{2n})=0$. 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 ${\bf R}^{2n}$ for a broad class of symplectic orthogonal matrix P.