The gap between near commutativity and almost commutativity in symplectic category

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2013, 20: 71-76. doi: 10.3934/era.2013.20.71

The gap between near commutativity and almost commutativity in symplectic category

Lev Buhovski ^1,

1.	School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978

Received March 2013 Revised June 2013 Published August 2013

Abstract
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On any closed symplectic manifold of dimension greater than $ 2 $, we construct a pair of smooth functions, such that on the one hand, the uniform norm of their Poisson bracket equals to $ 1 $, but on the other hand, this pair cannot be reasonably approximated (in the uniform norm) by a pair of Poisson commuting smooth functions. This comes in contrast with the dimension $ 2 $ case, where by a partial case of a result of Zapolsky [13], an opposite statement holds.

Keywords: Poisson bracket, pseudo- holomorphic curves., Poisson bracket invariants, rigidity.

Mathematics Subject Classification: Primary: 53D9.

Citation: Lev Buhovski. The gap between near commutativity and almost commutativity in symplectic category. Electronic Research Announcements, 2013, 20: 71-76. doi: 10.3934/era.2013.20.71

References:

[1]	L. Buhovsky, The $2/3$-convergence rate for the Poisson bracket, Geom. and Funct. Analysis, 19 (2010), 1620-1649. doi: 10.1007/s00039-010-0045-z.
[2]	L. Buhovsky, M. Entov and L. Polterovich, Poisson brackets and symplectic invariants, Selecta Mathematica, 18 (2012), 89-157. doi: 10.1007/s00029-011-0068-9.
[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, Comm. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.
[5]	M. Entov and L. Polterovich, $C^0$-rigidity of Poisson brackets, in "Proceedings of the Joint Summer Research Conference on Symplectic Topology and Measure-Preserving Dynamical Systems" (eds. A. Fathi, Y.-G. Oh and C. Viterbo), Contemporary Mathematics, 512, AMS, Providence, RI, (2010), 25-32. doi: 10.1090/conm/512.
[6]	M. Entov and L. Polterovich, $C^0$-rigidity of the double Poisson bracket, Int. Math. Res. Notices, (2009), 1134-1158.
[7]	M. Entov, L. Polterovich and D. Rosen, Poisson brackets, quasi-states and symplectic integrators, Discrete and Continuous Dynamical Systems, 28 (2010), 1455-1468. doi: 10.3934/dcds.2010.28.1455.
[8]	M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure and Applied Math. Quarterly, 3 (2007), 1037-1055.
[9]	M. B. Hastings, Making almost commuting matrices commute, Comm. Math. Phys., 291 (2009), 321-345. doi: 10.1007/s00220-009-0877-2.
[10]	C. Pearcy and A. Shields, Almost commuting matrices, J. Funct. Anal., 33 (1979), 332-338. doi: 10.1016/0022-1236(79)90071-5.
[11]	L. Polterovich, Symplectic geometry of quantum noise, arXiv:1206.3707, (2012).
[12]	F. Zapolsky, Quasi-states and the Poisson bracket on surfaces, J. Mod. Dyn., 1 (2007), 465-475. doi: 10.3934/jmd.2007.1.465.
[13]	F. Zapolsky, On almost Poisson commutativity in dimension two, Electron. Res. Announc. Math. Sci., 17 (2010), 155-160. doi: 10.3934/era.2010.17.155.

show all references

References:

[1]	L. Buhovsky, The $2/3$-convergence rate for the Poisson bracket, Geom. and Funct. Analysis, 19 (2010), 1620-1649. doi: 10.1007/s00039-010-0045-z.
[2]	L. Buhovsky, M. Entov and L. Polterovich, Poisson brackets and symplectic invariants, Selecta Mathematica, 18 (2012), 89-157. doi: 10.1007/s00029-011-0068-9.
[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, Comm. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.
[5]	M. Entov and L. Polterovich, $C^0$-rigidity of Poisson brackets, in "Proceedings of the Joint Summer Research Conference on Symplectic Topology and Measure-Preserving Dynamical Systems" (eds. A. Fathi, Y.-G. Oh and C. Viterbo), Contemporary Mathematics, 512, AMS, Providence, RI, (2010), 25-32. doi: 10.1090/conm/512.
[6]	M. Entov and L. Polterovich, $C^0$-rigidity of the double Poisson bracket, Int. Math. Res. Notices, (2009), 1134-1158.
[7]	M. Entov, L. Polterovich and D. Rosen, Poisson brackets, quasi-states and symplectic integrators, Discrete and Continuous Dynamical Systems, 28 (2010), 1455-1468. doi: 10.3934/dcds.2010.28.1455.
[8]	M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure and Applied Math. Quarterly, 3 (2007), 1037-1055.
[9]	M. B. Hastings, Making almost commuting matrices commute, Comm. Math. Phys., 291 (2009), 321-345. doi: 10.1007/s00220-009-0877-2.
[10]	C. Pearcy and A. Shields, Almost commuting matrices, J. Funct. Anal., 33 (1979), 332-338. doi: 10.1016/0022-1236(79)90071-5.
[11]	L. Polterovich, Symplectic geometry of quantum noise, arXiv:1206.3707, (2012).
[12]	F. Zapolsky, Quasi-states and the Poisson bracket on surfaces, J. Mod. Dyn., 1 (2007), 465-475. doi: 10.3934/jmd.2007.1.465.
[13]	F. Zapolsky, On almost Poisson commutativity in dimension two, Electron. Res. Announc. Math. Sci., 17 (2010), 155-160. doi: 10.3934/era.2010.17.155.

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