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
\(N \in \mathbb {N}\) all closed geodesics of index
\(\le \)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.,
$$\begin{aligned} \mathrm{ind}(c^\mathrm{rm})=m \,\mathrm{ind}(c^r)-(m-1)(n-1), \end{aligned}$$
for all
\(m \ge 1\) with
\(\mathrm{ind}\left( c^\mathrm{rm}\right) \le N.\) But this leads to a contradiction for
\(N =\infty \) 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
\(<L\) which do not intersect.