Add to ORKGIn 1973, Katok constructed a non-degenerate (also called bumpy) Finsler metric on $S^3$ with exactly four prime closed geodesics. And then Anosov conjectured that four should be the optimal lower bound of the number of prime closed geodesics on every Finsler $S^3$. In this paper, we proved this conjecture for bumpy Finsler $S^{3}$ if the Morse index of any prime closed geodesic is nonzero.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1504.07007, arXiv:1510.02872, arXiv:1508.0557
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