On almost Poisson commutativity in dimension two
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.
Received: June 2010; Revised: October 2010; Available Online: December 2010. |