On almost Poisson commutativity in dimension two

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

On almost Poisson commutativity in dimension two

Frol Zapolsky ^1,

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

Received June 2010 Revised October 2010 Published December 2010

Abstract
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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.

Keywords: Poisson bracket, surfaces..

Mathematics Subject Classification: Primary: 57M50; Secondary: 53D9.

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), 41-67. 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), 1620-1649. 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), 235-284. doi: 10.1215/00127094-2008-036.
[4]	M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.
[5]	M. Entov and L. Polterovich, ($C^0$)-rigidity of Poisson brackets, Symplectic topology and measure preserving dynamical systems, 25-32, Contemp. Math., 512, Amer. Math. Soc., Providence, RI, 2010.
[6]	M. Entov, L. Polterovich and D. Rosen, Poisson brackets, quasi-states and symplectic integrators, preprint, arXiv:0910.1980.
[7]	M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., Special Issue: In honor of Grigory Margulis. Part 1, 3 (2007), 1037-1055.
[8]	H. Federer, "Geometric Measure Theory," Die Grundl. der math. Wiss., vol. 153, Springer-Verlag New York Inc., New York 1969.
[9]	C. Pearcy and A. Shields, Almost commuting matrices, J. Funct. Anal., 33 (1979), 332-338. doi: 10.1016/0022-1236(79)90071-5.
[10]	F. Zapolsky, Quasi-states and the Poisson bracket on surfaces, J. Mod. Dyn., 1 (2007), 465-475.

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References:

[1]	J. F. Aarnes, Quasi-states and quasi-measures, Adv. Math., 86 (1991), 41-67. 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), 1620-1649. 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), 235-284. doi: 10.1215/00127094-2008-036.
[4]	M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.
[5]	M. Entov and L. Polterovich, ($C^0$)-rigidity of Poisson brackets, Symplectic topology and measure preserving dynamical systems, 25-32, Contemp. Math., 512, Amer. Math. Soc., Providence, RI, 2010.
[6]	M. Entov, L. Polterovich and D. Rosen, Poisson brackets, quasi-states and symplectic integrators, preprint, arXiv:0910.1980.
[7]	M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., Special Issue: In honor of Grigory Margulis. Part 1, 3 (2007), 1037-1055.
[8]	H. Federer, "Geometric Measure Theory," Die Grundl. der math. Wiss., vol. 153, Springer-Verlag New York Inc., New York 1969.
[9]	C. Pearcy and A. Shields, Almost commuting matrices, J. Funct. Anal., 33 (1979), 332-338. doi: 10.1016/0022-1236(79)90071-5.
[10]	F. Zapolsky, Quasi-states and the Poisson bracket on surfaces, J. Mod. Dyn., 1 (2007), 465-475.

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