On almost Poisson commutativity in dimension two
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Submission date: 13. Sep. 2010
published in: Electronic research announcements in mathematical sciences, 17 (2010), p. 155-160
DOI number (of the published article): 10.3934/era.2010.17.155
MSC-Numbers: 53D05, 58C35
Keywords and phrases: Poisson bracket, quasi-states, surfaces
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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.