Multiple P-cyclic symmetric closed characteristics on compact convex P-cyclic symmetric hypersurfaces in ℝ2n

Abstract

Let k ⩾ 2 be an integer and P be a 2n × 2n symplectic orthogonal matrix satisfying Pk = I2n and ker(Pj − I2n) = 0, 1 ⩽ j < k. For any compact convex hypersurface Σ ⊂ ℝ2n with n ⩾ 2 which is P-cyclic symmetric, i.e., x ∈ Σ implies Px ∈ Σ, we prove that if Σ is (r, R)-pinched with $$R/r<\sqrt{(2k+2)/k}$$ then there exist at least n geometrically distinct P-cyclic symmetric closed characteristics on Σ for a broad class of matrices P.