On continuity of quasi-morphisms for symplectic maps

TL;DR

This paper investigates the properties and structure of $C^0$-continuous homogeneous quasi-morphisms on symplectic groups, revealing their infinite-dimensional nature in certain cases and applications to Hofer's geometry.

Contribution

It demonstrates the infinite-dimensionality of $C^0$-continuous quasi-morphisms for specific symplectic manifolds and provides a topological characterization for surfaces, with applications to Hofer's geometry.

Findings

Infinite-dimensional space of quasi-morphisms for symplectic balls and surfaces (excluding the sphere).

Topological characterization of quasi-morphisms on surfaces.

Application to Hofer's geometry on Hamiltonian diffeomorphism groups.

Abstract

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.