A dichotomy for the Hofer growth of area preserving maps on the sphere via symmetrization

Abstract

We prove that autonomous Hamiltonian flows on the two-sphere exhibit the following dichotomy: the Hofer norm either grows linearly or is bounded in time by a universal constant C. Our approach involves a new technique, Hamiltonian symmetrization. Essentially, we prove that every autonomous Hamiltonian diffeomorphism is conjugate to an element C-close in the Hofer metric to one generated by a function of the height.

35 pages, 5 figures; revision of the exposition, added Proposition 2.16