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

TL;DR

This paper studies a special quasi-state on the 2D torus, calculates its values on Morse functions using Reeb graphs, and confirms its uniqueness, linking it to known quasi-morphisms.

Contribution

It proves the uniqueness of Grubb's quasi-state on the torus and computes its values on Morse functions via Reeb graphs, connecting to existing quasi-morphism results.

Findings

Confirmed the uniqueness of Grubb's quasi-state.

Derived a formula for the quasi-state on Morse functions.

Linked the quasi-state to quasi-morphisms on area-preserving diffeomorphisms.

Abstract

This note deals with quasi-states on the two-dimensional torus. Quasi-states 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 diffemorphisms, 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.