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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.

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