# American Institute of Mathematical Sciences

December  2010, 28(4): 1455-1468. doi: 10.3934/dcds.2010.28.1455

## Poisson brackets, quasi-states and symplectic integrators

 1 Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel 2 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Received  October 2009 Revised  February 2010 Published  June 2010

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.
Citation: Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455
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