Symplectic Geometry Seminar

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.