On almost Poisson commutativity in dimension two

Frol Zapolsky

doi:10.3934/era.2010.17.155

Article Contents

2010, Volume 17: 155-160. Doi: 10.3934/era.2010.17.155

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On almost Poisson commutativity in dimension two

Frol Zapolsky^1,

1.
Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig

Received: May 31, 2010

Revised: September 30, 2010

Published: December 2009

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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.

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