Multiplicity of closed geodesics on Finsler spheres with irrationally elliptic closed geodesics

If all prime closed geodesics on (Sn, F) with an irreversible Finsler metric F are irrationally elliptic, there exist either exactly 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {\frac{{n + 1}}{2}} \right]$$\end{document} or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler (S3, F) if any prime closed geodesic has non-zero Morse index.