On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus

TL;DR

This paper proves that the autonomous norm on the Hamiltonian diffeomorphism group of the 2D torus is unbounded, providing explicit examples and constructing quasimorphisms, including Calabi quasimorphisms.

Contribution

It establishes the unboundedness of the autonomous norm on the torus and constructs explicit examples and quasimorphisms demonstrating this property.

Findings

Autonomous norm on $Ham(T^2)$ is unbounded.

Explicit examples of Hamiltonian diffeomorphisms with large autonomous norm.

Construction of quasimorphisms, including Calabi quasimorphisms, on $Ham(T^2)$.

Abstract

We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide explicit examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct quasimorphisms on $Ham(T^2)$ and some of them are Calabi.