Two elliptic closed geodesics on positively curved Finsler spheres

Abstract

In this paper, we prove that for every Finsler $n$-dimensional sphere $(S^{n},F)$ with reversibility $\lm$ and flag curvature $K$ satisfying $\left(\frac{\lm}{1+\lm}\right)^2<K\le 1$, either there exist infinitely many closed geodesics, or there exist at least two elliptic closed geodesics and each linearized Poincar\'{e} map has at least one eigenvalue of the form $e^{\sqrt{-1}\th}$ with $\th$ being an irrational multiple of $\pi$.