Non-hyperbolic closed geodesics on Finsler spheres

Abstract

In this paper, we prove that for every bumpy Finsler $2k$-sphere $(S^{2k}, F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying the pinching condition $(\frac{\lambda}{\lambda + 1}) \lt K \leq 1$ either there exist infinitely many closed geodesics or there exist at least $2k$ non-hyperbolic closed geodesics. Due to the example of A. B. Katok, this estimate is sharp.