Hofer-Wysocki-Zehnder's Conjecture on Two or Infinitely Many Orbits

In their 2001 paper, Hofer, Wysocki and Zehnder conjectured that every autonomous Hamiltonian flow has either two or infinitely many simple periodic orbits on any compact star-shaped energy level; in the same paper, the authors prove this assuming in addition that the flow is nondegenerate and the stable and unstable manifolds of all hyperbolic orbits intersect transversally, a condition which holds generically. I will explain recent joint work resolving this conjecture. Our results also apply to show that every Finsler metric on the two-sphere has either two or infinitely many prime closed geodesics, answering a question attributed to Alvarez Paiva, Bangert and Long.