AI 中文总结
研究\({\bf R}^{2n}\)中星形超曲面上闭特征和不变集稳定性，通过比动态凸性弱的指标条件证闭特征非双曲，\(n = 3\)时在特定条件下证闭特征为椭圆，还在动态凸性下证明闭特征要么退化要么非局部极大。
AI 中文摘要
本文聚焦于\({\bf R}^{2n}\)中具有有限多个闭特征的星形超曲面上闭特征和不变集的稳定性。首先，在比动态凸性弱的某些指标条件下，证明所有闭特征是非双曲的。其次，当\(n = 3\)时，在非退化和一些小指标条件下，证明闭特征数恰好为\(3\)时它们都是椭圆的。最后，证明在动态凸性下所有闭特征要么是退化的，要么不是局部极大的。
英文摘要
This paper focuses on the stability of a closed characteristic and invariant sets on compact star shaped hypersurfaces in ${\bf R}^{2n}$ with finitely many simple closed characteristics. Firstly, it is proved that all closed characteristics are non-hyperbolic, under some index condition weaker than dynamical convexity. Secondly, when $n=3$, it is proved that all closed characteristics are elliptic when their number is exactly $3$, under non-degeneracy and some minor index condition. Lastly, it is proved that all closed characteristics are either degenerate at some iteration or not locally maximal under dynamical convexity.
