On Continuity of Quasimorphisms for Symplectic Maps

We discuss C-0-continuous homogeneous quasimorphisms on the identity component of the group of compactly supported symplectomorphisms of a symplectic manifold. Such quasimorphisms extend to the C-0-closure of this group inside the homeomorphism group. We show that for standard symplectic balls of any dimension, as well as for compact oriented surfaces other than the sphere, the space of such quasimorphisms is infinite-dimensional. In the case of surfaces, we give a user-friendly topological characterization of such quasimorphisms. We also present an application to Hofer's geometry on the group of Hamiltonian diffeomorphisms of the ball.