Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces

TL;DR

This paper develops new resonance identities for symmetric closed characteristics on symmetric star-shaped hypersurfaces and proves that if exactly two such characteristics exist in four dimensions, both are elliptic.

Contribution

It introduces two new resonance identities for symmetric closed characteristics and applies them to prove ellipticity of characteristics when only two exist in four dimensions.

Findings

Established two new resonance identities for symmetric closed characteristics.

Proved both characteristics are elliptic if exactly two exist in ${f R}^4$.

Extended previous identities to more general symmetric star-shaped hypersurfaces.

Abstract

So far, it is still unknown whether all the closed characteristics on a symmetric compact star-shaped hypersurface $\Sigma$ in ${\bf R}^{2n}$ are symmetric. In order to understand behaviors of such orbits, in this paper we establish first two new resonance identities for symmetric closed characteristics on symmetric compact star-shaped hypersurface $\Sigma$ in ${\bf R}^{2n}$ when there exist only finitely many geometrically distinct symmetric closed characteristics on $\Sigma$, which extend the identity established by Liu and Long in \cite{LLo1} of 2013 for symmetric strictly convex hypersurfaces. Then as an application of these identities and the identities established by Liu, Long and Wang recently in \cite{LLW1} for all closed characteristics on the same hypersurface, we prove that if there exist exactly two geometrically distinct closed characteristics on a symmetric compact star-shaped hypersuface in ${\bf R}^4$, then both of them must be elliptic.