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

In this paper, we prove that if all closed characteristics on a compact non-degenerate star-shaped hypersurface $Σ$ 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.

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