# American Institute of Mathematical Sciences

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

## The gap between near commutativity and almost commutativity in symplectic category

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

Received  March 2013 Revised  June 2013 Published  August 2013

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.
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. doi: 10.1007/s00039-010-0045-z. Google Scholar [2] L. Buhovsky, M. Entov and L. Polterovich, Poisson brackets and symplectic invariants,, Selecta Mathematica, 18 (2012), 89. doi: 10.1007/s00029-011-0068-9. Google Scholar [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. Google Scholar [4] M. Entov and L. Polterovich, Quasi-states and symplectic intersections,, Comm. Math. Helv., 81 (2006), 75. doi: 10.4171/CMH/43. Google Scholar [5] M. Entov and L. Polterovich, $C^0$-rigidity of Poisson brackets,, in, 512 (2010), 25. doi: 10.1090/conm/512. Google Scholar [6] M. Entov and L. Polterovich, $C^0$-rigidity of the double Poisson bracket,, Int. Math. Res. Notices, (2009), 1134. Google Scholar [7] M. Entov, L. Polterovich and D. Rosen, Poisson brackets, quasi-states and symplectic integrators,, Discrete and Continuous Dynamical Systems, 28 (2010), 1455. doi: 10.3934/dcds.2010.28.1455. Google Scholar [8] M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket,, Pure and Applied Math. Quarterly, 3 (2007), 1037. Google Scholar [9] M. B. Hastings, Making almost commuting matrices commute,, Comm. Math. Phys., 291 (2009), 321. doi: 10.1007/s00220-009-0877-2. Google Scholar [10] C. Pearcy and A. Shields, Almost commuting matrices,, J. Funct. Anal., 33 (1979), 332. doi: 10.1016/0022-1236(79)90071-5. Google Scholar [11] L. Polterovich, Symplectic geometry of quantum noise,, , (2012). Google Scholar [12] F. Zapolsky, Quasi-states and the Poisson bracket on surfaces,, J. Mod. Dyn., 1 (2007), 465. doi: 10.3934/jmd.2007.1.465. Google Scholar [13] F. Zapolsky, On almost Poisson commutativity in dimension two,, Electron. Res. Announc. Math. Sci., 17 (2010), 155. doi: 10.3934/era.2010.17.155. Google Scholar

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##### References:
 [1] L. Buhovsky, The $2/3$-convergence rate for the Poisson bracket,, Geom. and Funct. Analysis, 19 (2010), 1620. doi: 10.1007/s00039-010-0045-z. Google Scholar [2] L. Buhovsky, M. Entov and L. Polterovich, Poisson brackets and symplectic invariants,, Selecta Mathematica, 18 (2012), 89. doi: 10.1007/s00029-011-0068-9. Google Scholar [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. Google Scholar [4] M. Entov and L. Polterovich, Quasi-states and symplectic intersections,, Comm. Math. Helv., 81 (2006), 75. doi: 10.4171/CMH/43. Google Scholar [5] M. Entov and L. Polterovich, $C^0$-rigidity of Poisson brackets,, in, 512 (2010), 25. doi: 10.1090/conm/512. Google Scholar [6] M. Entov and L. Polterovich, $C^0$-rigidity of the double Poisson bracket,, Int. Math. Res. Notices, (2009), 1134. Google Scholar [7] M. Entov, L. Polterovich and D. Rosen, Poisson brackets, quasi-states and symplectic integrators,, Discrete and Continuous Dynamical Systems, 28 (2010), 1455. doi: 10.3934/dcds.2010.28.1455. Google Scholar [8] M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket,, Pure and Applied Math. Quarterly, 3 (2007), 1037. Google Scholar [9] M. B. Hastings, Making almost commuting matrices commute,, Comm. Math. Phys., 291 (2009), 321. doi: 10.1007/s00220-009-0877-2. Google Scholar [10] C. Pearcy and A. Shields, Almost commuting matrices,, J. Funct. Anal., 33 (1979), 332. doi: 10.1016/0022-1236(79)90071-5. Google Scholar [11] L. Polterovich, Symplectic geometry of quantum noise,, , (2012). Google Scholar [12] F. Zapolsky, Quasi-states and the Poisson bracket on surfaces,, J. Mod. Dyn., 1 (2007), 465. doi: 10.3934/jmd.2007.1.465. Google Scholar [13] F. Zapolsky, On almost Poisson commutativity in dimension two,, Electron. Res. Announc. Math. Sci., 17 (2010), 155. doi: 10.3934/era.2010.17.155. Google Scholar
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