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.

17 pages, to appear in Mathematische Zeitschrift. arXiv admin note: substantial text overlap with arXiv:2205.07082, arXiv:1510.08648, arXiv:1601.03470, arXiv:1405.5739, arXiv:1308.3904, arXiv:1308.3543