On Hofer's Geometry of Autonomous Flows on the Two-Sphere

The growth of an autonomous Hamiltonian flow in Hofer's metric is not yet well understood. A result of Polterovich and Rosen shows that generically this growth is asymptotically linear, and in all known cases where it is not, it appears to be bounded. Such a dichotomy is known for open connected surfaces of infinite area by a result of Polterovich and Siburg from 2000. I will discuss a new approach to this question, which establishes a strong version of such a dichotomy for the two-sphere. This talk is based on a joint work with Lev Buhovsky, Ben Feuerstein, and Leonid Polterovich.