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

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.