On almost Poisson commutativity in dimension two

TL;DR

This paper proves that on two-dimensional symplectic manifolds, functions with small Poisson brackets can be approximated by commuting functions in the uniform norm, with implications for function theory and quasi-states.

Contribution

It establishes a positive approximation result for almost commuting functions in dimension two, extending to manifolds with volume forms, using geometric measure theory techniques.

Findings

Functions with small Poisson brackets can be approximated by commuting functions in 2D.

The result applies to surfaces with area forms and volume forms.

Immediate applications to function theory and quasi-states on surfaces.

Abstract

Consider the following question: given two functions on a symplectic manifold whose Poisson bracket is small, is it possible to approximate them in the $C^0$ norm by commuting functions? We give a positive answer in dimension two, as a particular case of a more general statement which applies to functions on a manifold with a volume form. This result is based on a lemma in the spirit of geometric measure theory. We give some immediate applications to function theory and the theory of quasi-states on surfaces with area forms.