The gap between near commutativity and almost commutativity in symplectic category

TL;DR

This paper constructs examples on higher-dimensional symplectic manifolds showing a significant gap between near commutativity and almost commutativity of functions, contrasting with the 2D case where they are closer.

Contribution

It demonstrates that in symplectic geometry of dimension greater than 2, near commutativity does not imply approximation by commuting functions, revealing a fundamental difference from the 2D case.

Findings

Constructed pairs of functions with Poisson bracket norm 1 that cannot be approximated by commuting pairs.

Contrasts the higher-dimensional case with the 2D case where approximation is possible.

Highlights a dimension-dependent phenomenon in symplectic geometry.

Abstract

On any symplectic manifold of dimension greater than 2, we construct a pair of smooth functions, such that on the one hand, the uniform norm of their Poisson bracket equals to 1, but on the other hand, this pair cannot be reasonably approximated (in the uniform norm) by a pair of Poisson commuting smooth functions. This comes in contrast with the dimension 2 case, where by a partial case of a result of Zapolsky, an opposite statement holds.