摘要：

In this talk, we give a proof of the conjecture of Hofer-Wysocki-Zehnder published in 2003 asserting that a smooth and autonomous Hamiltonian flow on {\bf R}^4 has either two or infinitely many simple periodic orbits on any regular compact connected energy level that is transverse to the radial vector field. As its application, we also prove a longstanding conjecture of Bangert-Long that every irreversible Finsler metric on $S^2$ has either two or infinitely many prime closed geodesics. This talk is based on my recent joint work with Cristofaro-Gardiner, Hryniewicz and Hutchings.