Brake type closed characteristics on reversible compact convex hypersurfaces in View the MathML source

In this paper, let ΣΣ be a C3C3 compact convex in View the MathML sourceR2n satisfying the reversible condition NΣ=ΣNΣ=Σ with View the MathML sourceN=diag(−In,In). We prove that if there are exactly nn geometrically distinct closed characteristics on ΣΣ and all of them are nondegenerate, then all of them must be brake orbits up to a suitable translation of time. Moreover, for n=2n=2 or 3, we prove that if there are exactly nn geometrically distinct closed characteristics on ΣΣ, then all of them must be brake orbits up to a suitable translation of time.