# 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), 1620-1649. 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), 89-157. 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), 235-284. doi: 10.1215/00127094-2008-036.  Google Scholar [4] M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comm. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.  Google Scholar [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.  Google Scholar [6] M. Entov and L. Polterovich, $C^0$-rigidity of the double Poisson bracket, Int. Math. Res. Notices, (2009), 1134-1158.  Google Scholar [7] M. Entov, L. Polterovich and D. Rosen, Poisson brackets, quasi-states and symplectic integrators, Discrete and Continuous Dynamical Systems, 28 (2010), 1455-1468. 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), 1037-1055.  Google Scholar [9] M. B. Hastings, Making almost commuting matrices commute, Comm. Math. Phys., 291 (2009), 321-345. doi: 10.1007/s00220-009-0877-2.  Google Scholar [10] C. Pearcy and A. Shields, Almost commuting matrices, J. Funct. Anal., 33 (1979), 332-338. doi: 10.1016/0022-1236(79)90071-5.  Google Scholar [11] L. Polterovich, Symplectic geometry of quantum noise, arXiv:1206.3707, (2012). Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comm. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.  Google Scholar [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.  Google Scholar [6] M. Entov and L. Polterovich, $C^0$-rigidity of the double Poisson bracket, Int. Math. Res. Notices, (2009), 1134-1158.  Google Scholar [7] M. Entov, L. Polterovich and D. Rosen, Poisson brackets, quasi-states and symplectic integrators, Discrete and Continuous Dynamical Systems, 28 (2010), 1455-1468. 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), 1037-1055.  Google Scholar [9] M. B. Hastings, Making almost commuting matrices commute, Comm. Math. Phys., 291 (2009), 321-345. doi: 10.1007/s00220-009-0877-2.  Google Scholar [10] C. Pearcy and A. Shields, Almost commuting matrices, J. Funct. Anal., 33 (1979), 332-338. doi: 10.1016/0022-1236(79)90071-5.  Google Scholar [11] L. Polterovich, Symplectic geometry of quantum noise, arXiv:1206.3707, (2012). Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Sobhan Seyfaddini. Spectral killers and Poisson bracket invariants. Journal of Modern Dynamics, 2015, 9: 51-66. doi: 10.3934/jmd.2015.9.51 [2] Karina Samvelyan, Frol Zapolsky. Rigidity of the ${{L}^{p}}$-norm of the Poisson bracket on surfaces. Electronic Research Announcements, 2017, 24: 28-37. doi: 10.3934/era.2017.24.004 [3] Frol Zapolsky. Quasi-states and the Poisson bracket on surfaces. Journal of Modern Dynamics, 2007, 1 (3) : 465-475. doi: 10.3934/jmd.2007.1.465 [4] Francisco Crespo, Francisco Javier Molero, Sebastián Ferrer. Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier. Journal of Geometric Mechanics, 2016, 8 (2) : 169-178. doi: 10.3934/jgm.2016002 [5] Ermal Feleqi, Franco Rampazzo. Integral representations for bracket-generating multi-flows. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4345-4366. doi: 10.3934/dcds.2015.35.4345 [6] Nancy López Reyes, Luis E. Benítez Babilonia. A discrete hierarchy of double bracket equations and a class of negative power series. Mathematical Control & Related Fields, 2017, 7 (1) : 41-52. doi: 10.3934/mcrf.2017003 [7] Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847 [8] Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185 [9] Kaushik Nath, Palash Sarkar. Efficient arithmetic in (pseudo-)Mersenne prime order fields. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020113 [10] Boris Kolev. Poisson brackets in Hydrodynamics. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 555-574. doi: 10.3934/dcds.2007.19.555 [11] J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford and Masahico Saito. State-sum invariants of knotted curves and surfaces from quandle cohomology. Electronic Research Announcements, 1999, 5: 146-156. [12] Dmitry Tamarkin. Quantization of Poisson structures on R^2. Electronic Research Announcements, 1997, 3: 119-120. [13] C. Davini, F. Jourdan. Approximations of degree zero in the Poisson problem. Communications on Pure & Applied Analysis, 2005, 4 (2) : 267-281. doi: 10.3934/cpaa.2005.4.267 [14] Lubomir Kostal, Shigeru Shinomoto. Efficient information transfer by Poisson neurons. Mathematical Biosciences & Engineering, 2016, 13 (3) : 509-520. doi: 10.3934/mbe.2016004 [15] Oliver Knill. A deterministic displacement theorem for Poisson processes. Electronic Research Announcements, 1997, 3: 110-113. [16] Yacine Aït Amrane, Rafik Nasri, Ahmed Zeglaoui. Warped Poisson brackets on warped products. Journal of Geometric Mechanics, 2014, 6 (3) : 279-296. doi: 10.3934/jgm.2014.6.279 [17] Frol Zapolsky. On almost Poisson commutativity in dimension two. Electronic Research Announcements, 2010, 17: 155-160. doi: 10.3934/era.2010.17.155 [18] Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455 [19] Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051 [20] Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565

2020 Impact Factor: 0.929