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In this paper, we first generalize the common index jump theorem of Long-Zhuin 2002 and Duan-Long-Wang in 2016 to the case where the mean indices ofsymplectic paths are not required to be all positive. As applications, we studyclosed 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 distinctclosed characteristics on every compact non-degenerate perfect star-shapedhypersurface $\Sigma$ in ${\bf R}^{2n}$ provided that every prime closedcharacteristic possesses nonzero mean index. Furthermore, in the case of ${\bfR}^6$ we remove the nonzero mean index condition by showing that the existenceof only finitely many geometrically distinct closed characteristics impliesthat each of them must possess nonzero mean index. We also generalize the aboveresults about closed characteristics on non-degenerate star-shapedhypersurfaces to closed Reeb orbits of non-degenerate contact forms on a broadclass of prequantization bundles.

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