Reeb graph and quasi-states on the two-dimensional torus

This note deals with quasi-states on the two-dimensional torus. Quasistates are certain quasi-linear functionals (introduced by Aarnes) on the space of continuous functions. Grubb constructed a quasi-state on the torus, which is invariant under the group of area-preserving diffeomorphisms, and which moreover vanishes on functions having support in an open disk. Knudsen asserted the uniqueness of such a quasi-state; for the sake of completeness, we provide a proof. We calculate the value of Grubb’s quasi-state on Morse functions with distinct critical values via their Reeb graphs. The resulting formula coincides with the one obtained by Py in his work on quasi-morphisms on the group of area-preserving diffeomorphisms of the torus. Included is a short introduction to the link between quasi-states and quasi-morphisms in symplectic geometry.