Bumpy Metrics on Spheres and Minimal Index Growth

The existence of two geometrically distinct closed geodesics on an n -dimensional sphere S^n with a non-reversible and bumpy Finsler metric was shown independently by Duan and Long [ 7 ] and the author [ 25 ]. We simplify the proof of this statement by the following observation: If for some N ∈ℕ all closed geodesics of index ≤ N of a non-reversible and bumpy Finsler metric on S^n are geometrically equivalent to the closed geodesic c , then there is a covering c^r of minimal index growth, i.e., ind(c^rm)=m ind(c^r)-(m-1)(n-1), for all m ≥ 1 with ind( c^rm) ≤ N. But this leads to a contradiction for N =∞ as pointed out by Goresky and Hingston [ 13 ]. We also discuss perturbations of Katok metrics on spheres of even dimension carrying only finitely many closed geodesics. For arbitrarily large L>0 , we obtain on S^2 a metric of positive flag curvature carrying only two closed geodesics of length 更多