Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in $${\varvec{R}}^{2n}$$

Abstract

In this paper, let $$n$$ be a positive integer and $$P=diag(-I_{n-\kappa },I_\kappa ,-I_{n-\kappa },I_\kappa )$$ for some integer $$\kappa \in [0, n]$$ , we prove that for any compact convex hypersurface $$\Sigma $$ in $$\mathbf{R}^{2n}$$ with $$n\ge 2$$ there exist at least two geometrically distinct P-invariant closed characteristics on $$\Sigma $$ , provided that $$\Sigma $$ is P-symmetric, i.e., $$x\in \Sigma $$ implies $$Px\in \Sigma $$ . This work is shown to extend and unify several earlier works on this subject.