Closed geodesics on positively curved Finsler spheres

Abstract

In this paper, we prove that for every Finsler $n$-sphere $(S^n, F)$ for $n\ge 3$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda}{\lambda+1})^2<K\le 1$, either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincar\'e map has at least one eigenvalue which is of the form $\exp(\pi i \mu)$ with an irrational $\mu$. Furthermore, there always exist three prime closed geodesics on any $(S^3, F)$ satisfying the above pinching condition.