On continuity of quasi-morphisms for symplectic maps

Details On continuity of quasi-morphisms for symplectic maps On continuity of quasi-morphisms for symplectic maps

2009-04-08

arxiv.org/abs/0904.1397v1

Mathematics

Dynamical Systems

with an appendix by Michael Khanevsky

Scientific paper

We discuss $C^0$-continuous homogeneous quasi-morphisms on the identity component of the group of compactly supported symplectomorphisms of a symplectic manifold. Such quasi-morphisms 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 quasi-morphisms is infinite-dimensional. In the case of surfaces, we give a user-friendly topological characterization of such quasi-morphisms. We also present an application to Hofer's geometry on the group of Hamiltonian diffeomorphisms of the ball.

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