Poisson brackets, quasi-states and symplectic integrators

Details Poisson brackets, quasi-states and symplectic integrators Poisson brackets, quasi-states and symplectic integrators

2009-10-11

arxiv.org/abs/0910.1980v1

Mathematics

Symplectic Geometry

23 pages

Scientific paper

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.

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