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In this paper, we firstly generalize some theories developed by I. Ekelandand H. Hofer in [EkH] for closed characteristics on compact convexhypersurfaces in ${\bf R}^{2n}$ to star-shaped hypersurfaces. As applications,we use Ekeland-Hofer theory and index iteration theory to prove that if acompact star-shaped hypersuface in ${\bf R}^4$ satisfying some suitablepinching condition carries exactly two geometrically distinct closedcharacteristics, then both of them must be elliptic. We also conclude that thetheory developed by Y. Long and C. Zhu in [LoZ] still holds for dynamicallyconvex star-shaped hypersurfaces, and combining it with the results in [WHL],[LLW], [Wan3], we obtain that there exist at least $n$ closed characteristicson every dynamically convex star-shaped hypersurface in ${\bf R}^{2n}$ for$n=3, 4$.

Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in ${\bf R}^{2n}$