Electronic Research Announcements in Mathematical Sciences (ERA-MS)

On almost Poisson commutativity in dimension two

Pages: 155 - 160, 2010      doi:10.3934/era.2010.17.155

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Frol Zapolsky - Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany (email)

Abstract: Consider the following question: given two functions on a symplectic manifold whose Poisson bracket is small, is it possible to approximate them in the $C^0$ norm by commuting functions? We give a positive answer in dimension two, as a particular case of a more general statement which applies to functions on a manifold with a volume form. This result is based on a lemma in the spirit of geometric measure theory. We give some immediate applications to function theory and the theory of quasi-states on surfaces with area forms.

Keywords:  Poisson bracket, surfaces.
Mathematics Subject Classification:  Primary: 57M50; Secondary: 53D99.

Received: June 2010;      Revised: October 2010;      Available Online: December 2010.