zbMATH — the first resource for mathematics

Poisson brackets, quasi-states and symplectic integrators. (English) Zbl 1200.53068
Summary: 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.

53D05 Symplectic manifolds (general theory)
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
Full Text: DOI arXiv