Poisson brackets, quasi-states and symplectic integrators

Michael Entov; Leonid Polterovich; Daniel Rosen

doi:10.3934/dcds.2010.28.1455

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2010, Volume 28, Issue 4: 1455-1468. Doi: 10.3934/dcds.2010.28.1455

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Poisson brackets, quasi-states and symplectic integrators

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

Received: October 2009

Revised: February 2010

Published: December 2010

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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.

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Mathematics Subject Classification: Primary: 53D35, 37M15; Secondary: 65P10.

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