A dichotomy result for closed characteristics on compact star-shaped hypersurfaces in $\mathbf{R}^{2n}$

Abstract: In this paper, we prove that if all closed characteristics on a compact non-degenerate star-shaped hypersurface $\Sigma$ in $\mathbf{R}^{2n}$ are elliptic, then either there exist exactly $n$ geometrically distinct closed characteristics, or there exist infinitely many geometrically distinct closed characteristics.