Stability of closed characteristics and invariant sets on star-shaped hypersurfaces

Abstract

This paper focuses on the stability of a closed characteristic and invariant sets on compact star shaped hypersurfaces in ${\mathbf{R}}^{2n}$ 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.