The translation number and quasi-morphisms on groups of symplectomorphisms of the disk

TL;DR

This paper constructs two homogeneous quasi-morphisms on symplectomorphism groups of the disk, linking them to classical invariants and the Poincaré translation number, advancing understanding of symplectic group structures.

Contribution

It introduces two new homogeneous quasi-morphisms related to the Calabi invariant and flux, connecting them to the translation number on symplectomorphism groups.

Findings

Constructed two homogeneous quasi-morphisms on symplectomorphism groups.

Established relations between quasi-morphisms, Calabi invariant, and flux homomorphism.

Linked quasi-morphisms to the Poincaré translation number.

Abstract

On groups of symplectomorphisms of the disk, we construct two homogeneous quasi-morphisms which relate to the Calabi invariant and the flux homomorphism respectively. We also show the relation between the quasi-morphisms and the translation number introduced by Poincar\'{e}.