# American Institute of Mathematical Sciences

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

## On almost Poisson commutativity in dimension two

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

Received  June 2010 Revised  October 2010 Published  December 2010

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.
Citation: Frol Zapolsky. On almost Poisson commutativity in dimension two. Electronic Research Announcements, 2010, 17: 155-160. doi: 10.3934/era.2010.17.155
##### References:
 [1] J. F. Aarnes, Quasi-states and quasi-measures,, Adv. Math., 86 (1991), 41. doi: 10.1016/0001-8708(91)90035-6. [2] L. Buhovski, The 2/3 - convergence rate for the Poisson bracket,, Geom. Funct. Anal., 19 (2010), 1620. doi: 10.1007/s00039-010-0045-z. [3] F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations,, Duke Math. J., 144 (2008), 235. doi: 10.1215/00127094-2008-036. [4] M. Entov and L. Polterovich, Quasi-states and symplectic intersections,, Comment. Math. Helv., 81 (2006), 75. doi: 10.4171/CMH/43. [5] M. Entov and L. Polterovich, ($C^0$)-rigidity of Poisson brackets,, Symplectic topology and measure preserving dynamical systems, (2010), 25. [6] M. Entov, L. Polterovich and D. Rosen, Poisson brackets, quasi-states and symplectic integrators,, preprint, (). [7] M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket,, Pure Appl. Math. Q., 3 (2007), 1037. [8] H. Federer, "Geometric Measure Theory,", Die Grundl. der math. Wiss., 153 (1969). [9] C. Pearcy and A. Shields, Almost commuting matrices,, J. Funct. Anal., 33 (1979), 332. doi: 10.1016/0022-1236(79)90071-5. [10] F. Zapolsky, Quasi-states and the Poisson bracket on surfaces,, J. Mod. Dyn., 1 (2007), 465.

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##### References:
 [1] J. F. Aarnes, Quasi-states and quasi-measures,, Adv. Math., 86 (1991), 41. doi: 10.1016/0001-8708(91)90035-6. [2] L. Buhovski, The 2/3 - convergence rate for the Poisson bracket,, Geom. Funct. Anal., 19 (2010), 1620. doi: 10.1007/s00039-010-0045-z. [3] F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations,, Duke Math. J., 144 (2008), 235. doi: 10.1215/00127094-2008-036. [4] M. Entov and L. Polterovich, Quasi-states and symplectic intersections,, Comment. Math. Helv., 81 (2006), 75. doi: 10.4171/CMH/43. [5] M. Entov and L. Polterovich, ($C^0$)-rigidity of Poisson brackets,, Symplectic topology and measure preserving dynamical systems, (2010), 25. [6] M. Entov, L. Polterovich and D. Rosen, Poisson brackets, quasi-states and symplectic integrators,, preprint, (). [7] M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket,, Pure Appl. Math. Q., 3 (2007), 1037. [8] H. Federer, "Geometric Measure Theory,", Die Grundl. der math. Wiss., 153 (1969). [9] C. Pearcy and A. Shields, Almost commuting matrices,, J. Funct. Anal., 33 (1979), 332. doi: 10.1016/0022-1236(79)90071-5. [10] F. Zapolsky, Quasi-states and the Poisson bracket on surfaces,, J. Mod. Dyn., 1 (2007), 465.
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