Poisson brackets, quasi-states and symplectic integrators

TL;DR

This paper explores the rigidity phenomena related to Poisson brackets and symplectic quasi-states, extending bounds to iterated brackets and applying symplectic integrators to advance symplectic approximation theory.

Contribution

It extends existing bounds on Poisson brackets to iterated brackets using symplectic integrators and discusses applications and open problems in symplectic approximation.

Findings

Extended bounds to iterated Poisson brackets

Introduced symplectic integrators for rigidity analysis

Discussed applications to symplectic approximation theory

Abstract

This paper is a fusion of a survey and a research article. We focus on certain rigidity phenomena in function spaces associated to a symplectic manifold. Our starting point is a lower bound obtained in an earlier paper with Zapolsky for the uniform norm of the Poisson bracket of a pair of functions in terms of symplectic quasi-states. After a short review of the theory of symplectic quasi-states, we extend this bound to the case of iterated Poisson brackets. A new technical ingredient is the use of symplectic integrators. In addition, we discuss some applications to symplectic approximation theory and present a number of open problems.