Generalized common index jump theorem with applications to closed characteristics on star-shaped hypersurfaces and beyond

In this paper, we first generalize the common index jump theorem of Long-Zhu in 2002 and Duan-Long-Wang in 2016 to the case where the mean indices of symplectic paths are not required to be all positive. As applications, we study closed characteristics on compact star-shaped hypersurfaces in ${\bf R}^{2n}$, when both positive and negative mean indices may appear simultaneously. Specially we establish the existence of at least $n$ geometrically distinct closed characteristics on every compact non-degenerate perfect star-shaped hypersurface $\Sigma$ in ${\bf R}^{2n}$ provided that every prime closed characteristic possesses nonzero mean index. Furthermore, in the case of ${\bf R}^6$ we remove the nonzero mean index condition by showing that the existence of only finitely many geometrically distinct closed characteristics implies that each of them must possess nonzero mean index. We also generalize the above results about closed characteristics on non-degenerate star-shaped hypersurfaces to closed Reeb orbits of non-degenerate contact forms on a broad class of prequantization bundles.