On the minimal number of closed geodesics on positively curved Finsler spheres

In this paper, we proved that for every Finsler metric on Sⁿ (n≥ 4) with reversibility λ and flag curvature K satisfying (2n-3/n-1)² (λ/λ+1)²<K≤ 1 and λ<n-1/n-2, there exist at least n prime closed geodesics on (Sⁿ,F), which solved a conjecture of Katok and Anosov for such positivley curved spheres when n is even. Furthermore, if the number of closed geodesics on such positively curved Finsler Sⁿ is finite, then there exist at least 2[n/2]-1 non-hyperbolic closed geodesics.