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