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

الأصل · EN

So far, it is still unknown whether all the closed characteristics on a symmetric compact star-shaped hypersurface Σ in R²ⁿ 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 Σ in R²ⁿ when there exist only finitely many geometrically distinct symmetric closed characteristics on Σ, which extend the identity established by Liu and Long in 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 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 R⁴, then both of them must be elliptic.